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class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Ikhtisar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ikhtisar"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ikhtisar</span> </div> </a> <ul id="toc-Ikhtisar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi_dan_contoh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definisi_dan_contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definisi dan contoh</span> </div> </a> <button aria-controls="toc-Definisi_dan_contoh-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Definisi dan contoh</span> </button> <ul id="toc-Definisi_dan_contoh-sublist" class="vector-toc-list"> <li id="toc-Grup_Matriks_Lie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grup_Matriks_Lie"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Grup Matriks Lie</span> </div> </a> <ul id="toc-Grup_Matriks_Lie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konsep_terkait" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konsep_terkait"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Konsep terkait</span> </div> </a> <ul id="toc-Konsep_terkait-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definisi_topologi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definisi_topologi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Definisi topologi</span> </div> </a> <ul id="toc-Definisi_topologi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh_pertama" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contoh_pertama"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Contoh pertama</span> </div> </a> <ul id="toc-Contoh_pertama-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bukan_contoh" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bukan_contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Bukan contoh</span> </div> </a> <ul id="toc-Bukan_contoh-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lebih_banyak_contoh_dari_grup_Lie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lebih_banyak_contoh_dari_grup_Lie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Lebih banyak contoh dari grup Lie</span> </div> </a> <button aria-controls="toc-Lebih_banyak_contoh_dari_grup_Lie-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Lebih banyak contoh dari grup Lie</span> </button> <ul id="toc-Lebih_banyak_contoh_dari_grup_Lie-sublist" class="vector-toc-list"> <li id="toc-Dimensi_satu_dan_dua" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensi_satu_dan_dua"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Dimensi satu dan dua</span> </div> </a> <ul id="toc-Dimensi_satu_dan_dua-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Contoh_tambahan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Contoh_tambahan"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Contoh tambahan</span> </div> </a> <ul id="toc-Contoh_tambahan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konstruksi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konstruksi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Konstruksi</span> </div> </a> <ul id="toc-Konstruksi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pengertian_terkait" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pengertian_terkait"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Pengertian terkait</span> </div> </a> <ul id="toc-Pengertian_terkait-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Konsep_dasar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konsep_dasar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Konsep dasar</span> </div> </a> <button aria-controls="toc-Konsep_dasar-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Konsep dasar</span> </button> <ul id="toc-Konsep_dasar-sublist" class="vector-toc-list"> <li id="toc-Peta_eksponensial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Peta_eksponensial"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Peta eksponensial</span> </div> </a> <ul id="toc-Peta_eksponensial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgrup_Lie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subgrup_Lie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Subgrup Lie</span> </div> </a> <ul id="toc-Subgrup_Lie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Wakilan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wakilan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Wakilan</span> </div> </a> <ul id="toc-Wakilan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sejarah_awal" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah_awal"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Sejarah awal</span> </div> </a> <ul id="toc-Sejarah_awal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Konsep_grup_Lie,_dan_kemungkinan_klasifikasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Konsep_grup_Lie,_dan_kemungkinan_klasifikasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Konsep grup Lie, dan kemungkinan klasifikasi</span> </div> </a> <ul id="toc-Konsep_grup_Lie,_dan_kemungkinan_klasifikasi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Catatan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Catatan"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Catatan</span> </div> </a> <button aria-controls="toc-Catatan-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Catatan</span> </button> <ul id="toc-Catatan-sublist" class="vector-toc-list"> <li id="toc-Catatan_penjelasan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Catatan_penjelasan"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Catatan penjelasan</span> </div> </a> <ul id="toc-Catatan_penjelasan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kutipan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kutipan"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Kutipan</span> </div> </a> <ul id="toc-Kutipan-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Gulingkan daftar isi" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Grup Lie</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Pergi ke artikel dalam bahasa lain. Terdapat 36 bahasa" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">36 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D9%84%D9%8A" title="زمرة لي – Arab" lang="ar" hreflang="ar" data-title="زمرة لي" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_%D0%9B%D1%96" title="Група Лі – Belarusia" lang="be" hreflang="be" data-title="Група Лі" data-language-autonym="Беларуская" data-language-local-name="Belarusia" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_de_Lie" title="Grup de Lie – Katalan" lang="ca" hreflang="ca" data-title="Grup de Lie" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Lieova_grupa" title="Lieova grupa – Cheska" lang="cs" hreflang="cs" data-title="Lieova grupa" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Liegruppe" title="Liegruppe – Dansk" lang="da" hreflang="da" data-title="Liegruppe" data-language-autonym="Dansk" data-language-local-name="Dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lie-Gruppe" title="Lie-Gruppe – Jerman" lang="de" hreflang="de" data-title="Lie-Gruppe" data-language-autonym="Deutsch" data-language-local-name="Jerman" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Lie_group" title="Lie group – Inggris" lang="en" hreflang="en" data-title="Lie group" data-language-autonym="English" data-language-local-name="Inggris" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo_de_Lie" title="Grupo de Lie – Esperanto" lang="eo" hreflang="eo" data-title="Grupo de Lie" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_de_Lie" title="Grupo de Lie – Spanyol" lang="es" hreflang="es" data-title="Grupo de Lie" data-language-autonym="Español" data-language-local-name="Spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_%D9%84%DB%8C" title="گروه لی – Persia" lang="fa" hreflang="fa" data-title="گروه لی" data-language-autonym="فارسی" data-language-local-name="Persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lien_ryhm%C3%A4" title="Lien ryhmä – Suomi" lang="fi" hreflang="fi" data-title="Lien ryhmä" data-language-autonym="Suomi" data-language-local-name="Suomi" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_de_Lie" title="Groupe de Lie – Prancis" lang="fr" hreflang="fr" data-title="Groupe de Lie" data-language-autonym="Français" data-language-local-name="Prancis" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%9C%D7%99" title="חבורת לי – Ibrani" lang="he" hreflang="he" data-title="חבורת לי" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Lie-csoport" title="Lie-csoport – Hungaria" lang="hu" hreflang="hu" data-title="Lie-csoport" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_de_Lie" title="Gruppo de Lie – Interlingua" lang="ia" hreflang="ia" data-title="Gruppo de Lie" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_di_Lie" title="Gruppo di Lie – Italia" lang="it" hreflang="it" data-title="Gruppo di Lie" data-language-autonym="Italiano" data-language-local-name="Italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E7%BE%A4" title="リー群 – Jepang" lang="ja" hreflang="ja" data-title="リー群" data-language-autonym="日本語" data-language-local-name="Jepang" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A6%AC_%EA%B5%B0" title="리 군 – Korea" lang="ko" hreflang="ko" data-title="리 군" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Lie-Grupp" title="Lie-Grupp – Luksemburg" lang="lb" hreflang="lb" data-title="Lie-Grupp" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luksemburg" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lie-groep" title="Lie-groep – Belanda" lang="nl" hreflang="nl" data-title="Lie-groep" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Lie-gruppe" title="Lie-gruppe – Bokmål Norwegia" lang="nb" hreflang="nb" data-title="Lie-gruppe" data-language-autonym="Norsk bokmål" data-language-local-name="Bokmål Norwegia" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B2%E0%A8%BE%E0%A8%88_%E0%A8%97%E0%A8%B0%E0%A9%81%E0%A9%B1%E0%A8%AA" title="ਲਾਈ ਗਰੁੱਪ – Punjabi" lang="pa" hreflang="pa" data-title="ਲਾਈ ਗਰੁੱਪ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_Liego" title="Grupa Liego – Polski" lang="pl" hreflang="pl" data-title="Grupa Liego" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_de_Lie" title="Grupo de Lie – Portugis" lang="pt" hreflang="pt" data-title="Grupo de Lie" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Grup_Lie" title="Grup Lie – Rumania" lang="ro" hreflang="ro" data-title="Grup Lie" data-language-autonym="Română" data-language-local-name="Rumania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%9B%D0%B8" title="Группа Ли – Rusia" lang="ru" hreflang="ru" data-title="Группа Ли" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Liova_grupa" title="Liova grupa – Slovak" lang="sk" hreflang="sk" data-title="Liova grupa" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Liejeva_grupa" title="Liejeva grupa – Sloven" lang="sl" hreflang="sl" data-title="Liejeva grupa" data-language-autonym="Slovenščina" data-language-local-name="Sloven" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B8%D1%98%D0%B5%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Лијева група – Serbia" lang="sr" hreflang="sr" data-title="Лијева група" data-language-autonym="Српски / srpski" data-language-local-name="Serbia" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Liegrupp" title="Liegrupp – Swedia" lang="sv" hreflang="sv" data-title="Liegrupp" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D3%AF%D2%B3%D0%B8_%D0%9B%D0%B8" title="Гурӯҳи Ли – Tajik" lang="tg" hreflang="tg" data-title="Гурӯҳи Ли" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Lie_grubu" title="Lie grubu – Turki" lang="tr" hreflang="tr" data-title="Lie grubu" data-language-autonym="Türkçe" data-language-local-name="Turki" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_%D0%9B%D1%96" title="Група Лі – Ukraina" lang="uk" hreflang="uk" data-title="Група Лі" data-language-autonym="Українська" data-language-local-name="Ukraina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_Lie" title="Nhóm Lie – Vietnam" lang="vi" hreflang="vi" data-title="Nhóm Lie" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9D%8E%E7%BE%A4" title="李群 – Tionghoa" lang="zh" hreflang="zh" data-title="李群" data-language-autonym="中文" data-language-local-name="Tionghoa" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9D%8E%E7%BE%A3" title="李羣 – Kanton" lang="yue" hreflang="yue" data-title="李羣" data-language-autonym="粵語" data-language-local-name="Kanton" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q622679#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div 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semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Grup_Lie" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa" title="Daftar semua halaman istimewa [q]" accesskey="q"><span>Halaman istimewa</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Grup_Lie&oldid=25294171" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Grup_Lie&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Grup_Lie&id=25294171&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FGrup_Lie"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrKodu&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FGrup_Lie"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Grup+Lie"><span>Buat buku</span></a></li><li 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ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><p><br /> </p> <style data-mw-deduplicate="TemplateStyles:r26333518">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 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3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse"><tbody><tr><th class="sidebar-title"><a class="mw-selflink selflink">Grup Lie</a></th></tr><tr><td class="sidebar-image" style="padding-bottom:0.9em;"><span typeof="mw:File/Frameless"><a href="/wiki/Berkas:E8Petrie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png" decoding="async" width="180" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/270px-E8Petrie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/360px-E8Petrie.svg.png 2x" data-file-width="2852" data-file-height="2863" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/w/index.php?title=Grup_klasik&action=edit&redlink=1" class="new" title="Grup klasik (halaman belum tersedia)">Grup klasik</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">Linear umum</a> GL(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_linear_khusus&action=edit&redlink=1" class="new" title="Grup linear khusus (halaman belum tersedia)">Linear khusus</a> LK(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_ortogonal&action=edit&redlink=1" class="new" title="Grup ortogonal (halaman belum tersedia)">Ortogonal</a> O(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_ortogonal_khusus&action=edit&redlink=1" class="new" title="Grup ortogonal khusus (halaman belum tersedia)">Ortogonal khusus</a> OK(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_uniter&action=edit&redlink=1" class="new" title="Grup uniter (halaman belum tersedia)">Uniter</a> U(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_uniter_khusus&action=edit&redlink=1" class="new" title="Grup uniter khusus (halaman belum tersedia)">Uniter khusus</a> UK(<i>n</i>)</li> <li><a href="/w/index.php?title=Grup_simplektik&action=edit&redlink=1" class="new" title="Grup simplektik (halaman belum tersedia)">Simplektik</a> Sp(<i>n</i>)</li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">Grup Lie sederhana</a></div><div class="sidebar-list-content mw-collapsible-content"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar hlist" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Klasik</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">A<sub><i>n</i></sub></a></li> <li><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">B<sub><i>n</i></sub></a></li> <li><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">C<sub><i>n</i></sub></a></li> <li><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">D<sub><i>n</i></sub></a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Eksepsional</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=G2_(matematika)&action=edit&redlink=1" class="new" title="G2 (matematika) (halaman belum tersedia)">G<sub>2</sub></a></li> <li><a href="/w/index.php?title=F4_(matematika)&action=edit&redlink=1" class="new" title="F4 (matematika) (halaman belum tersedia)">F<sub>4</sub></a></li> <li><a href="/w/index.php?title=E6_(matematika)&action=edit&redlink=1" class="new" title="E6 (matematika) (halaman belum tersedia)">E<sub>6</sub></a></li> <li><a href="/w/index.php?title=E7_(matematika)&action=edit&redlink=1" class="new" title="E7 (matematika) (halaman belum tersedia)">E<sub>7</sub></a></li> <li><a href="/w/index.php?title=E8_(matematika)&action=edit&redlink=1" class="new" title="E8 (matematika) (halaman belum tersedia)">E<sub>8</sub></a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/w/index.php?title=Tabel_Grup_Lie&action=edit&redlink=1" class="new" title="Tabel Grup Lie (halaman belum tersedia)">Grup Lie lainnya</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">Lingkaran</a></li> <li><a href="/w/index.php?title=Grup_Lorentz&action=edit&redlink=1" class="new" title="Grup Lorentz (halaman belum tersedia)">Lorentz</a></li> <li><a href="/w/index.php?title=Grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Grup Poincaré (halaman belum tersedia)"> Poincaré</a></li> <li><a href="/w/index.php?title=Grup_konformal&action=edit&redlink=1" class="new" title="Grup konformal (halaman belum tersedia)">Grup konformal</a></li> <li><a href="/w/index.php?title=Difeomorfisme&action=edit&redlink=1" class="new" title="Difeomorfisme (halaman belum tersedia)">Difeomorfisme</a></li> <li><a href="/w/index.php?title=Grup_loop&action=edit&redlink=1" class="new" title="Grup loop (halaman belum tersedia)">Loop</a></li> <li><a href="/w/index.php?title=Grup_Euklides&action=edit&redlink=1" class="new" title="Grup Euklides (halaman belum tersedia)">Euklides</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/wiki/Aljabar_Lie" title="Aljabar Lie">Aljabar Lie</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/w/index.php?title=Korespondensi_grup_Lie%E2%80%93aljabar_Lie&action=edit&redlink=1" class="new" title="Korespondensi grup Lie–aljabar Lie (halaman belum tersedia)">Korespondensi grup Lie–aljabar Lie</a></li> <li><a href="/w/index.php?title=Peta_eksponensial_(teori_Lie)&action=edit&redlink=1" class="new" title="Peta eksponensial (teori Lie) (halaman belum tersedia)">Peta eksponensial</a></li> <li><a href="/w/index.php?title=Representasi_adjoin&action=edit&redlink=1" class="new" title="Representasi adjoin (halaman belum tersedia)">Representasi adjoin</a></li> <li><style data-mw-deduplicate="TemplateStyles:r23782733">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"><ul><li><a href="/w/index.php?title=Bentuk_Killing&action=edit&redlink=1" class="new" title="Bentuk Killing (halaman belum tersedia)">Bentuk Killing</a></li><li><a href="/w/index.php?title=Indeks_aljabar_Lie&action=edit&redlink=1" class="new" title="Indeks aljabar Lie (halaman belum tersedia)">Indeks</a></li></ul></div></li> <li><a href="/w/index.php?title=Simetri_titik_Lie&action=edit&redlink=1" class="new" title="Simetri titik Lie (halaman belum tersedia)">Simetri titik Lie</a></li> <li><a href="/w/index.php?title=Aljabar_Lie_sederhana&action=edit&redlink=1" class="new" title="Aljabar Lie sederhana (halaman belum tersedia)">Aljabar Lie sederhana</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/w/index.php?title=Aljabar_Lie_semisederhana&action=edit&redlink=1" class="new" title="Aljabar Lie semisederhana (halaman belum tersedia)">Aljabar Lie semisederhana</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/w/index.php?title=Diagram_Dynkin&action=edit&redlink=1" class="new" title="Diagram Dynkin (halaman belum tersedia)">Diagram Dynkin</a></li> <li><a href="/w/index.php?title=Subaljabar_Cartan&action=edit&redlink=1" class="new" title="Subaljabar Cartan (halaman belum tersedia)">Subaljabar Cartan</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><ul><li><a href="/w/index.php?title=Sistem_akar&action=edit&redlink=1" class="new" title="Sistem akar (halaman belum tersedia)">Sistem akar</a></li><li><a href="/w/index.php?title=Grup_Weyl&action=edit&redlink=1" class="new" title="Grup Weyl (halaman belum tersedia)">Grup Weyl</a></li></ul></div></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><ul><li><a href="/w/index.php?title=Bentuk_riil_(teori_Lie)&action=edit&redlink=1" class="new" title="Bentuk riil (teori Lie) (halaman belum tersedia)">Bentuk riil</a></li><li><a href="/w/index.php?title=Kompleksifikasi_(grup_Lie)&action=edit&redlink=1" class="new" title="Kompleksifikasi (grup Lie) (halaman belum tersedia)">Kompleksifikasi</a></li></ul></div></li> <li><a href="/w/index.php?title=Aljabar_Lie_slip&action=edit&redlink=1" class="new" title="Aljabar Lie slip (halaman belum tersedia)">Aljabar Lie slip</a></li> <li><a href="/w/index.php?title=Aljabar_Lie_kompak&action=edit&redlink=1" class="new" title="Aljabar Lie kompak (halaman belum tersedia)">Aljabar Lie kompak</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title"><a href="/w/index.php?title=Teori_wakilan&action=edit&redlink=1" class="new" title="Teori wakilan (halaman belum tersedia)">Teori wakilan</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/w/index.php?title=Wakilan_dari_grup_Lie&action=edit&redlink=1" class="new" title="Wakilan dari grup Lie (halaman belum tersedia)">Representasi grup Lie</a></li> <li><a href="/w/index.php?title=Wakilan_aljabar_Lie&action=edit&redlink=1" class="new" title="Wakilan aljabar Lie (halaman belum tersedia)">Wakilan aljabar Lie</a></li> <li><a href="/w/index.php?title=Teori_wakilan_dari_aljabar_Lie_semisederhana&action=edit&redlink=1" class="new" title="Teori wakilan dari aljabar Lie semisederhana (halaman belum tersedia)">Teori wakilan dari aljabar Lie semisederhana</a></li> <li><a href="/w/index.php?title=Wakilan_dari_grup_Lie_klasik&action=edit&redlink=1" class="new" title="Wakilan dari grup Lie klasik (halaman belum tersedia)">Wakilan dari grup Lie klasik</a></li> <li><a href="/w/index.php?title=Teorema_bobot_tertinggi&action=edit&redlink=1" class="new" title="Teorema bobot tertinggi (halaman belum tersedia)">Teorema bobot tertinggi</a></li> <li><a href="/w/index.php?title=Teorema_Borel%E2%80%93Weil%E2%80%93_Bott&action=edit&redlink=1" class="new" title="Teorema Borel–Weil– Bott (halaman belum tersedia)">Teorema Borel–Weil– Bott</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title">Grup Lie dalam <a href="/wiki/Physics" class="mw-redirect" title="Physics">physics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/w/index.php?title=Fisika_partikel_dan_teori_repsentasi&action=edit&redlink=1" class="new" title="Fisika partikel dan teori repsentasi (halaman belum tersedia)">Fisika partikel dan teori repsentasi</a></li> <li><a href="/w/index.php?title=Teori_representasi_dari_grup_Lorentz&action=edit&redlink=1" class="new" title="Teori representasi dari grup Lorentz (halaman belum tersedia)">Representasi grup Lorentz</a></li> <li><a href="/w/index.php?title=Teori_representasi_grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Teori representasi grup Poincaré (halaman belum tersedia)">Representasi grup Poincaré</a></li> <li><a href="/w/index.php?title=Teori_representasi_dari_geru0_Galilea&action=edit&redlink=1" class="new" title="Teori representasi dari geru0 Galilea (halaman belum tersedia)">Representasi grup Galilea</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title">Ilmuwan</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li> <li><a href="/w/index.php?title=Claude_Chevalley&action=edit&redlink=1" class="new" title="Claude Chevalley (halaman belum tersedia)">Claude Chevalley</a></li> <li><a href="/w/index.php?title=Harish-Chandra&action=edit&redlink=1" class="new" title="Harish-Chandra (halaman belum tersedia)">Harish-Chandra</a></li> <li><a href="/w/index.php?title=Armand_Borel&action=edit&redlink=1" class="new" title="Armand Borel (halaman belum tersedia)">Armand Borel</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below plainlist"> <ul><li><a href="/w/index.php?title=Glosarium_grup_Lie_dan_aljabar_Lie&action=edit&redlink=1" class="new" title="Glosarium grup Lie dan aljabar Lie (halaman belum tersedia)">Glosarium</a></li> <li><a href="/w/index.php?title=Tabel_grup_Lie&action=edit&redlink=1" class="new" title="Tabel grup Lie (halaman belum tersedia)">Tabel grup Lie</a></li></ul></td></tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r18590415">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Grup_Lie" title="Templat:Grup Lie"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Grup_Lie" title="Pembicaraan Templat:Grup Lie"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Grup_Lie&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r26333525">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}html.client-js body.skin-minerva .mw-parser-output .mbox-text-span{margin-left:23px!important}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Periksaterjemahan plainlinks metadata ambox ambox-content ambox-rough_translation" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/Berkas:Translation_to_english_arrow.svg" class="mw-file-description" title="Translation arrow icon"><img alt="Translation arrow icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/50px-Translation_to_english_arrow.svg.png" decoding="async" width="50" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/75px-Translation_to_english_arrow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Translation_to_english_arrow.svg/100px-Translation_to_english_arrow.svg.png 2x" data-file-width="60" data-file-height="20" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Artikel atau sebagian dari artikel ini mungkin diterjemahkan dari <i><a href="https://en.wikipedia.org/wiki/Lie_group" class="extiw" title="en:Lie group">Lie group</a></i> di en.wikipedia.org. <b>Isinya masih belum akurat</b>, karena bagian yang diterjemahkan masih perlu diperhalus dan disempurnakan. Jika Anda menguasai bahasa aslinya, harap pertimbangkan untuk menelusuri referensinya dan menyempurnakan terjemahan ini. Anda juga dapat ikut bergotong royong pada <a href="/wiki/Wikipedia:ProyekWiki_Perbaikan_Terjemahan" title="Wikipedia:ProyekWiki Perbaikan Terjemahan">ProyekWiki Perbaikan Terjemahan</a>.<br /> <small>(Pesan ini dapat dihapus jika terjemahan dirasa sudah cukup tepat. Lihat pula: <a href="/wiki/Wikipedia:Panduan_dalam_menerjemahkan_artikel" title="Wikipedia:Panduan dalam menerjemahkan artikel">panduan penerjemahan artikel</a>)</small></div></td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar sidebar-collapse" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Struktur_aljabar" title="Struktur aljabar">Struktur aljabar</a> → <b>Teori grup</b></span><br /><a href="/wiki/Teori_grup" title="Teori grup">Teori grup</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/Berkas:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;">Gagasan dasar</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgrup" title="Subgrup">Subgrup</a></li> <li><a href="/wiki/Subgrup_normal" title="Subgrup normal">Subgrup normal</a></li></ul> <ul><li><a href="/wiki/Grup_hasil_bagi" title="Grup hasil bagi">Grup hasil bagi</a></li> <li><a href="/w/index.php?title=Grup_darab_langsung&action=edit&redlink=1" class="new" title="Grup darab langsung (halaman belum tersedia)">darab langsung</a></li> <li><a href="/w/index.php?title=Grup_semi-darab_langsung&action=edit&redlink=1" class="new" title="Grup semi-darab langsung (halaman belum tersedia)">semi-darab langsung</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Homomorfisme_grup" class="mw-redirect" title="Homomorfisme grup">Homomorfisme grup</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(aljabar)#Homomorfisme_grup" title="Kernel (aljabar)">kernel</a></li> <li><a href="/wiki/Bayangan_(matematika)" title="Bayangan (matematika)">bayangan</a></li> <li><a href="/w/index.php?title=Jumlah_grup_langsung&action=edit&redlink=1" class="new" title="Jumlah grup langsung (halaman belum tersedia)">jumlah langsung</a></li></ul> <ul><li><a href="/w/index.php?title=Darab_karangan_bunga&action=edit&redlink=1" class="new" title="Darab karangan bunga (halaman belum tersedia)">karangan bunga</a></li> <li><a href="/wiki/Grup_sederhana" title="Grup sederhana">sederhana</a></li> <li><a href="/wiki/Grup_hingga" title="Grup hingga">hingga</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_takhingga&action=edit&redlink=1" class="new" title="Grup takhingga (halaman belum tersedia)">takhingga</a></li> <li><a href="/w/index.php?title=Grup_kontinu&action=edit&redlink=1" class="new" title="Grup kontinu (halaman belum tersedia)">kontinu</a></li> <li><a href="/w/index.php?title=Grup_multiplikatif&action=edit&redlink=1" class="new" title="Grup multiplikatif (halaman belum tersedia)">multiplikatif</a></li></ul> <ul><li><a href="/wiki/Grup_aditif" title="Grup aditif">aditif</a></li> <li><a href="/wiki/Grup_siklik" title="Grup siklik">siklik</a></li> <li><a href="/w/index.php?title=Grup_Abel&action=edit&redlink=1" class="new" title="Grup Abel (halaman belum tersedia)">Abel</a></li> <li><a href="/wiki/Grup_dihedral" title="Grup dihedral">dihedral</a></li></ul> <ul><li><a href="/wiki/Grup_nilpoten" title="Grup nilpoten">nilpoten</a></li> <li><a href="/w/index.php?title=Grup_terselesaikan&action=edit&redlink=1" class="new" title="Grup terselesaikan (halaman belum tersedia)">terselesaikan</a></li> <li><a href="/w/index.php?title=Aksi_grup&action=edit&redlink=1" class="new" title="Aksi grup (halaman belum tersedia)">aksi</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=Glosarium_teori_grup&action=edit&redlink=1" class="new" title="Glosarium teori grup (halaman belum tersedia)">Glosarium teori grup</a></li> <li><a href="/wiki/Daftar_topik_teori_grup" title="Daftar topik teori grup">Daftar topik teori grup</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Grup_hingga" title="Grup hingga">Grup hingga</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Klasifikasi_grup_sederhana_hingga" title="Klasifikasi grup sederhana hingga">Klasifikasi grup sederhana hingga</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Grup_siklik" title="Grup siklik">siklik</a></li> <li><a href="/w/index.php?title=Grup_bergantian&action=edit&redlink=1" class="new" title="Grup bergantian (halaman belum tersedia)">bergantian</a></li> <li><a href="/w/index.php?title=Grup_tipe_Lie&action=edit&redlink=1" class="new" title="Grup tipe Lie (halaman belum tersedia)">tipe Lie</a></li> <li><a href="/w/index.php?title=Grup_sporadik&action=edit&redlink=1" class="new" title="Grup sporadik (halaman belum tersedia)">sporadik</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=Teorema_Cauchy_(teori_grup)&action=edit&redlink=1" class="new" title="Teorema Cauchy (teori grup) (halaman belum tersedia)">Teorema Cauchy</a></li> <li><a href="/wiki/Teorema_Lagrange_(teori_grup)" title="Teorema Lagrange (teori grup)">Teorema Lagrange</a></li></ul> <ul><li><a href="/wiki/Teorema_Sylow" title="Teorema Sylow">Teorema Sylow</a></li> <li><a href="/w/index.php?title=Subgrup_Hall&action=edit&redlink=1" class="new" title="Subgrup Hall (halaman belum tersedia)">Teorema Hall</a></li></ul> <ul><li><a href="/wiki/Grup-p" title="Grup-p">grup-p</a></li> <li><a href="/w/index.php?title=Grup_Abel_elementer&action=edit&redlink=1" class="new" title="Grup Abel elementer (halaman belum tersedia)">Grup Abel elementer</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_Frobenius&action=edit&redlink=1" class="new" title="Grup Frobenius (halaman belum tersedia)">Grup Frobenius</a></li></ul> <ul><li><a href="/w/index.php?title=Pengganda_Schur&action=edit&redlink=1" class="new" title="Pengganda Schur (halaman belum tersedia)">Pengganda Schur</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Grup_simetrik" title="Grup simetrik">Grup simetrik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f98267cda84a9c8abfee60f7184af3cb1aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{n}}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_Klein&action=edit&redlink=1" class="new" title="Grup Klein (halaman belum tersedia)">Grup Klein</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664209da7650f00b3507efe25f89aeff9783146c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {V} }"></span></li> <li><a href="/wiki/Grup_dihedral" title="Grup dihedral">Grup dihedral</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8c7d71e610ba30a0856fa881290ae80b7282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{n}}"></span></li> <li><a href="/wiki/Grup_kuaternion" title="Grup kuaternion">Grup kuaternion</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ea6b6e5d15ac13060c9724fdbf3aa79b353f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathrm {Q} }"></span></li> <li><a href="/wiki/Grup_disiklik" title="Grup disiklik">Grup disiklik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Dic} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Dic} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911cef2c1b11010e151d3737a8f3b8588f3b00e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.673ex; height:2.509ex;" alt="{\displaystyle \mathrm {Dic} _{n}}"></span></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><ul><li><a href="/w/index.php?title=Grup_diskret&action=edit&redlink=1" class="new" title="Grup diskret (halaman belum tersedia)">Grup diskret</a></li><li><a href="/w/index.php?title=Kekisi_(subgrup_diskret)&action=edit&redlink=1" class="new" title="Kekisi (subgrup diskret) (halaman belum tersedia)">Kekisi</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Bilangan_bulat" title="Bilangan bulat">Bilangan bulat</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Grup_bebas" title="Grup bebas">Grup bebas</a></li></ul> <div style="padding:0.2em 0.4em; line-height:1.2em;"><a href="/w/index.php?title=Grup_modular&action=edit&redlink=1" class="new" title="Grup modular (halaman belum tersedia)">Grup modular</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f4d0e8493b732b05e29613a714405de8b25356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.884ex; height:2.843ex;" alt="{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}"></span></li><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe790f9dda6d5d14bf20e9f5f92d9b0ff83e696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.301ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbb {Z} )}"></span></li></ul></div></div> <ul><li><a href="/w/index.php?title=Grup_aritmetika&action=edit&redlink=1" class="new" title="Grup aritmetika (halaman belum tersedia)">Grup aritmetika</a></li> <li><a href="/w/index.php?title=Kekisi_(grup)&action=edit&redlink=1" class="new" title="Kekisi (grup) (halaman belum tersedia)">Kekisi</a></li> <li><a href="/w/index.php?title=Grup_hiperbolik&action=edit&redlink=1" class="new" title="Grup hiperbolik (halaman belum tersedia)">Grup hiperbolik</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/w/index.php?title=Grup_topologis&action=edit&redlink=1" class="new" title="Grup topologis (halaman belum tersedia)">Topologis</a> dan <a class="mw-selflink selflink">Grup Lie</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/w/index.php?title=Solenoid_(matematika)&action=edit&redlink=1" class="new" title="Solenoid (matematika) (halaman belum tersedia)">Solenoid</a></li> <li><a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">Lingkaran</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">Linear umum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba99fb253ee3b9082e5d718da746260073e6c7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.481ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_linear_khusus&action=edit&redlink=1" class="new" title="Grup linear khusus (halaman belum tersedia)">Linear khusus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SL} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cd269a9439a1075bb460bf6ae7b1407086c35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.949ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_ortogonal&action=edit&redlink=1" class="new" title="Grup ortogonal (halaman belum tersedia)">Ortogonal</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {O} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1471779b64c8868583dcd50e3c6381293f0dd67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_Euklides&action=edit&redlink=1" class="new" title="Grup Euklides (halaman belum tersedia)">Euklides</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {E} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f699fb0fd3801a34a5016afa897a24953fa9aab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.787ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_ortogonal_khusus&action=edit&redlink=1" class="new" title="Grup ortogonal khusus (halaman belum tersedia)">Ortogonal khusus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa71842f19b6810b4bfa9eb282e92fbf285094e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_uner&action=edit&redlink=1" class="new" title="Grup uner (halaman belum tersedia)">Uner</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {U} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {U} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32fa84df5de5dfa91b6bdc88fb03fc8792c9f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathrm {U} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_uniter_khusus&action=edit&redlink=1" class="new" title="Grup uniter khusus (halaman belum tersedia)">Uniter khusus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a205091aabd5690efdfeb7354a55844f2eb31b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_simplektik&action=edit&redlink=1" class="new" title="Grup simplektik (halaman belum tersedia)">Simplektik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Sp} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Sp} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00b37fb88bcb5053f70e6386aa2119328bd1171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \mathrm {Sp} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=G2_(matematika)&action=edit&redlink=1" class="new" title="G2 (matematika) (halaman belum tersedia)">G<sub>2</sub></a></li> <li><a href="/w/index.php?title=F4_(matematika)&action=edit&redlink=1" class="new" title="F4 (matematika) (halaman belum tersedia)">F<sub>4</sub></a></li> <li><a href="/w/index.php?title=E6_(matematika)&action=edit&redlink=1" class="new" title="E6 (matematika) (halaman belum tersedia)">E<sub>6</sub></a></li> <li><a href="/w/index.php?title=E7_(matematika)&action=edit&redlink=1" class="new" title="E7 (matematika) (halaman belum tersedia)">E<sub>7</sub></a></li> <li><a href="/w/index.php?title=E8_(matematika)&action=edit&redlink=1" class="new" title="E8 (matematika) (halaman belum tersedia)">E<sub>8</sub></a></li></ul> <ul><li><a href="/w/index.php?title=Grup_Loretnz&action=edit&redlink=1" class="new" title="Grup Loretnz (halaman belum tersedia)">Lorentz</a></li> <li><a href="/w/index.php?title=Grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Grup Poincaré (halaman belum tersedia)">Poincaré</a></li> <li><a href="/w/index.php?title=Group_konformal&action=edit&redlink=1" class="new" title="Group konformal (halaman belum tersedia)">konformal</a></li></ul> <ul><li><a href="/w/index.php?title=Difeomorfisme&action=edit&redlink=1" class="new" title="Difeomorfisme (halaman belum tersedia)">Difeomorfisme</a></li> <li><a href="/w/index.php?title=Grup_gelung&action=edit&redlink=1" class="new" title="Grup gelung (halaman belum tersedia)">Gelung</a></li></ul> <div style="padding:0.2em 0.4em; line-height:1.2em;"><a href="/w/index.php?title=Grup_Lie_berdimensi_takhingga&action=edit&redlink=1" class="new" title="Grup Lie berdimensi takhingga (halaman belum tersedia)">Grup Lie berdimensi takhingga</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b333ad74141a87280c1fbe4ae31d7bc0dcc572a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.906ex; height:2.843ex;" alt="{\displaystyle O(\infty )}"></span></li><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SU} (\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c73a9f1ca8515559b3a55fd76827f11457af591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.169ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (\infty )}"></span></li><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Sp} (\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Sp} (\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92382a0fb0bdd299850c5505365353df0b04a921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle \mathrm {Sp} (\infty )}"></span></li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/w/index.php?title=Grup_aljabar&action=edit&redlink=1" class="new" title="Grup aljabar (halaman belum tersedia)">Grup aljabar</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/w/index.php?title=Grup_aljabar_linear&action=edit&redlink=1" class="new" title="Grup aljabar linear (halaman belum tersedia)">Grup aljabar linear</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_reduktif&action=edit&redlink=1" class="new" title="Grup reduktif (halaman belum tersedia)">Grup reduktif</a></li></ul> <ul><li><a href="/w/index.php?title=Varietas_Abel&action=edit&redlink=1" class="new" title="Varietas Abel (halaman belum tersedia)">Varietas Abel</a></li></ul> <ul><li><a href="/wiki/Kurva_eliptik" title="Kurva eliptik">Kurva eliptik</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18590415"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Teori_grup_sidebar" title="Templat:Teori grup sidebar"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Teori_grup_sidebar" title="Pembicaraan Templat:Teori grup sidebar"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Teori_grup_sidebar&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r18844875">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Artikel ini bukan mengenai <a href="/w/index.php?title=Grup_tipe_Lie&action=edit&redlink=1" class="new" title="Grup tipe Lie (halaman belum tersedia)">Grup tipe Lie</a>.</div> <p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, <b>grup Lie</b> (<span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt"><a href="/wiki/Bantuan:IPA_untuk_bahasa_Inggris" class="mw-redirect" title="Bantuan:IPA untuk bahasa Inggris">/<span style="border-bottom:1px dotted"><span title="'l' in 'lie'">l</span><span title="/iː/: 'ee' in 'fleece'">iː</span></span>/</a></span></span> "Lee") adalah <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a> yang merupakan <a href="/w/index.php?title=Lipatan_berjenis&action=edit&redlink=1" class="new" title="Lipatan berjenis (halaman belum tersedia)">lipatan berjenis</a>. <a href="/wiki/Lipatan" class="mw-redirect" title="Lipatan">Lipatan</a> adalah ruang lokal <a href="/wiki/Ruang_Euklides" title="Ruang Euklides">ruang Euklides</a>, sedangkan grup mendefinisikan abstrak, konsep umum perkalian dan pengambilan invers (pembagian). Menggabungkan dua ide ini, kita akan mendapatkan <a href="/w/index.php?title=Grup_kontinu&action=edit&redlink=1" class="new" title="Grup kontinu (halaman belum tersedia)">grup kontinu</a> dimana poin dikalikan secara kebersamaan dan kebalikannya dapat diambil. Jika, sebagai penambahan, perkalian, dan pengambilan invers didefinisikan sebagai halus (terdiferensiasi), maka kita mendapatkan rumus grup Lie. </p><p>Grup Lie diberikan sebuah model alami untuk konsep <a href="/w/index.php?title=Simetri_kontinu&action=edit&redlink=1" class="new" title="Simetri kontinu (halaman belum tersedia)">simetri kontinu</a>, contohnya adalah simetri rotasi dalam tiga dimensi (diberikan oleh <a href="/w/index.php?title=Grup_ortogonal_khusus&action=edit&redlink=1" class="new" title="Grup ortogonal khusus (halaman belum tersedia)">grup ortogonal khusus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SO</mtext> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76ea569dfd0b7e75a686925ff43ea07e50e79526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(3)}"></span>). Grup Lie sering digunakan di banyak bagian matematika dan <a href="/wiki/Fisika" title="Fisika">fisika</a> modern. </p><p>Grup Lie pertama kali ditemukan dengan mempelajari subgrup <a href="/wiki/Matriks_(matematika)" title="Matriks (matematika)">matriks</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/097d8981d4fa62fb4f4d6353c450b5c844d89dc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.983ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}_{n}(\mathbb {R} )}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0303318438cf5afd77739497e1f3517c9e52d5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.983ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}_{n}(\mathbb {C} )}"></span>, <a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">grup dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matriks inver</a> di atas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> atau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. Ini disebut sebagai <a href="/w/index.php?title=Grup_klasik&action=edit&redlink=1" class="new" title="Grup klasik (halaman belum tersedia)">grup klasik</a>, karena konsepnya telah diperluas jauh melampaui asal-usulnya. Grup Lie dinamai menurut matematikawan asal Norwegia yaitu <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> (1842–1899) yang memberikan dasar teori <a href="/w/index.php?title=Grup_transformasi&action=edit&redlink=1" class="new" title="Grup transformasi (halaman belum tersedia)">grup transformasi</a> kontinu. Motivasi asli Lie untuk memperkenalkan grup Lie adalah untuk model kesimetrian kontinu dengan <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a> yang sama bahwa grup hingga digunakan dalam <a href="/wiki/Teori_Galois" title="Teori Galois">teori Galois</a> untuk model simetri diskrit <a href="/wiki/Persamaan_aljabar" title="Persamaan aljabar">persamaan aljabar</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ikhtisar">Ikhtisar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=1" title="Sunting bagian: Ikhtisar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=1" title="Sunting kode sumber bagian: Ikhtisar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Circle_as_Lie_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circle_as_Lie_group.svg/220px-Circle_as_Lie_group.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circle_as_Lie_group.svg/330px-Circle_as_Lie_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Circle_as_Lie_group.svg/440px-Circle_as_Lie_group.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Himpunan semua <a href="/wiki/Bilangan_kompleks" title="Bilangan kompleks">bilangan kompleks</a> dengan <a href="/wiki/Nilai_absolut" title="Nilai absolut">nilai absolut</a> 1 (terkait dengan titik-titik pada <a href="/wiki/Lingkaran" title="Lingkaran">lingkaran</a> dari pusat 0 dan jari-jari 1 di <a href="/w/index.php?title=Medan_kompleks&action=edit&redlink=1" class="new" title="Medan kompleks (halaman belum tersedia)">medan kompleks</a>) adalah grup Lie dalam perkalian kompleks: <a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">grup lingkaran</a>.</figcaption></figure> <p>Grup Lie adalah <a href="/w/index.php?title=Lipatan_berjenis&action=edit&redlink=1" class="new" title="Lipatan berjenis (halaman belum tersedia)">lipatan berjenis</a> <a href="/w/index.php?title=Kehalusan&action=edit&redlink=1" class="new" title="Kehalusan (halaman belum tersedia)">halus</a> dan dengan demikian dapat dipelajari menggunakan <a href="/wiki/Kalkulus_diferensial" title="Kalkulus diferensial">kalkulus diferensial</a> berbeda dengan <a href="/wiki/Grup_topologi" title="Grup topologi">grup topologi</a> umum. Salah satu ide kunci dalam teori grup Lie adalah mengganti objek <i>global</i> grup dengan versi <i>lokal</i> atau linierisasi. Grup Lie sendiri disebut sebagai "grup infinitesimal" dan dikenal sebagai <a href="/wiki/Aljabar_Lie" title="Aljabar Lie">aljabar Lie</a>. </p><p>Grup Lie memainkan peran yang sangat besar dalam <a href="/wiki/Geometri" title="Geometri">geometri</a> modern unruk beberapa tingkatan yang berbeda. <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> berpendapat dalam <a href="/w/index.php?title=Program_Erlangen&action=edit&redlink=1" class="new" title="Program Erlangen (halaman belum tersedia)">program Erlangen</a> dapat mempertimbangkan berbagai "geometri "dengan menentukan grup transformasi yang sesuai yang menghilangkan sifat geometris <a href="/w/index.php?title=Invarian_(matematika)&action=edit&redlink=1" class="new" title="Invarian (matematika) (halaman belum tersedia)">invarian</a>. Jadi <a href="/wiki/Geometri_Euklides" title="Geometri Euklides">geometri Euklides</a> dengan pilihan grup <a href="/w/index.php?title=Grup_Euklides&action=edit&redlink=1" class="new" title="Grup Euklides (halaman belum tersedia)">E(3)</a> dari transformasi jarak ruang Euklides <b>R</b><sup>3</sup> <a href="/w/index.php?title=Konformal_geometri&action=edit&redlink=1" class="new" title="Konformal geometri (halaman belum tersedia)">konformal geometri</a> dengan memperbesar grup ke <a href="/w/index.php?title=Grup_konformal&action=edit&redlink=1" class="new" title="Grup konformal (halaman belum tersedia)">grup konformal</a>, sedangkan dalam <a href="/wiki/Geometri_proyektif" title="Geometri proyektif">geometri proyektif</a> tertarik pada sifat invarian di bawah <a href="/w/index.php?title=Grup_proyektif&action=edit&redlink=1" class="new" title="Grup proyektif (halaman belum tersedia)">grup proyektif</a>. Ide ini kemudian mengarah pada gagasan tentang sebuah <a href="/w/index.php?title=Struktur-G&action=edit&redlink=1" class="new" title="Struktur-G (halaman belum tersedia)">struktur-G</a>, dimana <i>G</i> adalah grup Lie dari simetris "lokal" dari lipatan. </p><p>Grup Lie dan aljabar Lie memainkan peran utama dalam fisika modern, dengan grup Lie biasanya memainkan peran sebagai simetri sistem fisik. Di sini, <a href="/w/index.php?title=Grup_Lie_wakilan&action=edit&redlink=1" class="new" title="Grup Lie wakilan (halaman belum tersedia)">wakilan</a> dari grup Lie atau <a href="/w/index.php?title=Aljabar_Lie_wakilan&action=edit&redlink=1" class="new" title="Aljabar Lie wakilan (halaman belum tersedia)">aljabar Lie</a> sangat penting untuk penggunaannya. Teori representasi <a href="/w/index.php?title=Fisika_partikel_dan_teori_wakilan&action=edit&redlink=1" class="new" title="Fisika partikel dan teori wakilan (halaman belum tersedia)">digunakan secara luas dalam fisika partikel</a>. Grup wakilannya sangat penting untuk digunakan <a href="/w/index.php?title=Grup_rotasi_3D&action=edit&redlink=1" class="new" title="Grup rotasi 3D (halaman belum tersedia)">grup rotasi S(3)</a> atau <a href="/w/index.php?title=Grup_rotasi_3D&action=edit&redlink=1" class="new" title="Grup rotasi 3D (halaman belum tersedia)">penutup ganda SU(2)</a>, <a href="/w/index.php?title=Koefisien_Clebsch%E2%80%93Gordan_untuk_SU(3)&action=edit&redlink=1" class="new" title="Koefisien Clebsch–Gordan untuk SU(3) (halaman belum tersedia)">grup satuan khusus SU(3)</a> dan <a href="/w/index.php?title=Teori_wakilan_dalam_grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Teori wakilan dalam grup Poincaré (halaman belum tersedia)">grup Poincaré</a>. </p><p>Pada tingkat "global", setiap grup Lie <a href="/w/index.php?title=Grup_aksi_(matematika)&action=edit&redlink=1" class="new" title="Grup aksi (matematika) (halaman belum tersedia)">aksi</a> pada objek geometris, yaitu <a href="/w/index.php?title=Lipatan_Riemannian&action=edit&redlink=1" class="new" title="Lipatan Riemannian (halaman belum tersedia)">Riemannian</a> atau <a href="/w/index.php?title=Lipatan_simplektis&action=edit&redlink=1" class="new" title="Lipatan simplektis (halaman belum tersedia)">lipatan simplektis</a>, aksi ini memberikan ukuran dan menghasilkan <a href="/wiki/Struktur_aljabar" title="Struktur aljabar">struktur aljabar</a> yang banyak. Adanya simetri kontinu yang diekspresikan melalui <a href="/w/index.php?title=Grup_Lie_aksi&action=edit&redlink=1" class="new" title="Grup Lie aksi (halaman belum tersedia)">grup Lie aksi</a> pada lipatan menempatkan batasan yang kuat pada geometrinya dan memfasilitasi <a href="/w/index.php?title=Analisis_global&action=edit&redlink=1" class="new" title="Analisis global (halaman belum tersedia)">analisis</a> pada lipatan. Grup Lie aksi sangat penting dalam penggunaannya, dan dipelajari dalam <a href="/w/index.php?title=Teori_wakilan&action=edit&redlink=1" class="new" title="Teori wakilan (halaman belum tersedia)">teori wakilan</a>. </p><p>Pada 1940-an-1950-an, <a href="/w/index.php?title=Ellis_Kolchin&action=edit&redlink=1" class="new" title="Ellis Kolchin (halaman belum tersedia)">Ellis Kolchin</a>, <a href="/w/index.php?title=Armand_Borel&action=edit&redlink=1" class="new" title="Armand Borel (halaman belum tersedia)">Armand Borel</a>, dan <a href="/w/index.php?title=Claude_Chevalley&action=edit&redlink=1" class="new" title="Claude Chevalley (halaman belum tersedia)">Claude Chevalley</a> menyadari bahwa banyak hasil dasar mengenai grup Lie yang dikembangkan sepenuhnya secara aljabar sebagai teori <a href="/w/index.php?title=Grup_aljabar&action=edit&redlink=1" class="new" title="Grup aljabar (halaman belum tersedia)">grup aljabar</a> yang ditentukan melalui sembarang <a href="/wiki/Medan_(matematika)" class="mw-redirect" title="Medan (matematika)">medan</a>. Wawasan ini membuka kemungkinan baru dalam aljabar murni, dengan memberikan konstruksi seragam untuk sebagian besar <a href="/w/index.php?title=Grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Grup sederhana hingga (halaman belum tersedia)">grup sederhana hingga</a> serta dalam <a href="/wiki/Geometri_aljabar" title="Geometri aljabar">geometri aljabar</a>. Teori <a href="/w/index.php?title=Bentuk_automorfik&action=edit&redlink=1" class="new" title="Bentuk automorfik (halaman belum tersedia)">bentuk automorfik</a>, cabang penting dari <a href="/wiki/Teori_bilangan" title="Teori bilangan">teori bilangan</a> modern, berurusan secara ekstensif dengan analogi grup Lie selama <a href="/w/index.php?title=Gelanggang_Adele&action=edit&redlink=1" class="new" title="Gelanggang Adele (halaman belum tersedia)">gelanggang Adele</a>; <a href="/w/index.php?title=Bilangan_p-adik&action=edit&redlink=1" class="new" title="Bilangan p-adik (halaman belum tersedia)">bilangan p-adik</a> grup Lie memainkan peran penting dengan melalui koneksi dengan representasi Galois dalam teori bilangan. </p> <div class="mw-heading mw-heading2"><h2 id="Definisi_dan_contoh">Definisi dan contoh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=2" title="Sunting bagian: Definisi dan contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=2" title="Sunting kode sumber bagian: Definisi dan contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Grup Lie riil</b> adalah <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a> merupakan berdimensi riil hingga <a href="/w/index.php?title=Lipatan_berjenis&action=edit&redlink=1" class="new" title="Lipatan berjenis (halaman belum tersedia)">lipatan halus</a>, dimana operasi grup <a href="/wiki/Perkalian" title="Perkalian">perkalian</a> dan inversi adalah <a href="/w/index.php?title=Peta_halus&action=edit&redlink=1" class="new" title="Peta halus (halaman belum tersedia)">peta halus</a>. Maka perkalian grup, adalah </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>:</mo> <mi>G</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mspace width="1em" /> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd01d7b7887c62349a3b93dcd7baa71bf508fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.91ex; height:2.843ex;" alt="{\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy}"></span></dd></dl> <p>jadi <i>μ</i> adalah pemetaan halus dari <a href="/wiki/Lipatan#Produk_Kartesius" class="mw-redirect" title="Lipatan">produk berjenis</a> <span class="nowrap"><i>G</i> × <i>G</i></span> sebagai <i>G</i>. Kedua persyaratan ini dapat digabungkan menjadi satu persyaratan yaitu pemetaan </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)\mapsto x^{-1}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto x^{-1}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/502b30691789028f56d96adf7d94a576bddc5f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.76ex; height:3.176ex;" alt="{\displaystyle (x,y)\mapsto x^{-1}y}"></span></dd></dl> <p>sebagai pemetaan mulus dari produk berjenis yaitu <i>G</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Grup_Matriks_Lie">Grup Matriks Lie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=3" title="Sunting bagian: Grup Matriks Lie" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=3" title="Sunting kode sumber bagian: Grup Matriks Lie"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span> sebagai grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matriks invers dengan entri dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. <a href="/w/index.php?title=Teorema_subgrup_tertutup&action=edit&redlink=1" class="new" title="Teorema subgrup tertutup (halaman belum tersedia)">Subgrup tertutup</a> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span> adalah grup Lie<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> yang disebut <i>matriks grup Lie</i> Karena sebagian besar contoh dari grup Lie direalisasikan sebagai matriks grup Lie, beberapa buku teks membatasi perhatian pada kelas ini, termasuk yang ada dalam Hall<sup id="cite_ref-Hall_2-0" class="reference"><a href="#cite_note-Hall-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> dan Rossmann.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Membatasi sebuah matriks grup Lie dengan cara menyederhanakan definisi aljabar Lie dan peta eksponensial. Berikut ini adalah contoh standar grup matriks Lie. </p> <ul><li><a href="/w/index.php?title=Grup_linear_khusus&action=edit&redlink=1" class="new" title="Grup linear khusus (halaman belum tersedia)">Grup linear khusus</a> di atas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> yaitu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} (n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} (n,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24a96aa98eb162511808745d2a75af537de3600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.661ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (n,\mathbb {R} )}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afb5de04b309c90be458fd3bbf0b53ccb7e47c1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.661ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (n,\mathbb {C} )}"></span> terdiri dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matriks dengan determinan satu dan entri dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> atau <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span></li> <li><a href="/w/index.php?title=Grup_unital&action=edit&redlink=1" class="new" title="Grup unital (halaman belum tersedia)">Grup unital</a> dan grup uniter khusus yaitu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{U}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>U</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{U}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8003032ac7853dca278584b21d4e0fa51151cea7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle {\text{U}}(n)}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fea546d5a1fc1581b203f2ab1efc3118006f4263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(n)}"></span>, terdiri dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matriks kompleks <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U^{*}=U^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{*}=U^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2afc1f611c7a3f18e460394bb6506c763340b694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.168ex; height:2.676ex;" alt="{\displaystyle U^{*}=U^{-1}}"></span> (dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(U)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(U)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d43ccf95b37b676f5d386b240cb721b812a0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.082ex; height:2.843ex;" alt="{\displaystyle \det(U)=1}"></span> dalam kasus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fea546d5a1fc1581b203f2ab1efc3118006f4263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(n)}"></span>)</li> <li><a href="/w/index.php?title=Grup_ortogonal&action=edit&redlink=1" class="new" title="Grup ortogonal (halaman belum tersedia)">Grup ortogonal</a> dan grup ortogonal khusus yaitu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{O}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>O</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{O}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/702926ecf22593fa2c82e6ba19fcb3d9f3efd6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle {\text{O}}(n)}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SO</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db598c97edbf4abcc58bac4407621fce5382fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(n)}"></span>, terdiri dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"></span> matriks <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\mathrm {T} }=R^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{\mathrm {T} }=R^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b974469915f67080aa9ac2f96d6c566931008e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.378ex; height:2.676ex;" alt="{\displaystyle R^{\mathrm {T} }=R^{-1}}"></span> (dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(R)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(R)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef4e4006d672c548a3bd9975f097cb817e5a87c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.064ex; height:2.843ex;" alt="{\displaystyle \det(R)=1}"></span> dalam kasus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SO}}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SO</mtext> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SO}}(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db598c97edbf4abcc58bac4407621fce5382fae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle {\text{SO}}(n)}"></span>)</li></ul> <p>Semua contoh sebelumnya termasuk dalam tajuk <a href="/w/index.php?title=Grup_klasik&action=edit&redlink=1" class="new" title="Grup klasik (halaman belum tersedia)">grup klasik</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Konsep_terkait">Konsep terkait</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=4" title="Sunting bagian: Konsep terkait" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=4" title="Sunting kode sumber bagian: Konsep terkait"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/w/index.php?title=Grup_Lie_kompleks&action=edit&redlink=1" class="new" title="Grup Lie kompleks (halaman belum tersedia)">Grup Lie kompleks</a></b> didefinisikan dengan cara yang sama menggunakan <a href="/w/index.php?title=Lipatan_kompleks&action=edit&redlink=1" class="new" title="Lipatan kompleks (halaman belum tersedia)">lipatan kompleks</a> yang sebenarnya (contoh: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} (2,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} (2,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba7cdf4373aac3a21444fceecdd1101b9dd947a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.429ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} (2,\mathbb {C} )}"></span>), dan menggunakan alternatif <a href="/wiki/Ruang_metrik_lengkap#Pelengkap" title="Ruang metrik lengkap">pelengkap metrik</a> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>, grup topologi dimana setiap titik memiliki lingkungan <i>p</i>-adik. </p><p><a href="/w/index.php?title=Masalah_kelima_Hilbert&action=edit&redlink=1" class="new" title="Masalah kelima Hilbert (halaman belum tersedia)">Masalah kelima Hilbert</a> menanyakan apakah untuk mengganti lipatan yang dibedakan dengan topologi atau analitik dapat menghasilkan contoh baru. Jawaban atas pertanyaan ini ternyata negatif: pada tahun 1952 matematikawan <a href="/w/index.php?title=Andrew_Gleason&action=edit&redlink=1" class="new" title="Andrew Gleason (halaman belum tersedia)">Gleason</a>, <a href="/w/index.php?title=Deane_Montgomery&action=edit&redlink=1" class="new" title="Deane Montgomery (halaman belum tersedia)">Montgomery</a> dan <a href="/w/index.php?title=Leo_Zippin&action=edit&redlink=1" class="new" title="Leo Zippin (halaman belum tersedia)">Zippin</a> menunjukkan bahwa jika <i>G</i> adalah lipatan topologi, maka tepat satu struktur analitik pada <i>G</i> yang mengubah menjadi grup Lie (lihat pula <a href="/w/index.php?title=Konjektur_Hilbert%E2%80%93Smith&action=edit&redlink=1" class="new" title="Konjektur Hilbert–Smith (halaman belum tersedia)">Konjektur Hilbert–Smith</a>). Jika lipatan dasar yang berdimensi tak hingga (misalnya, <a href="/w/index.php?title=Lipatan_Hilbert&action=edit&redlink=1" class="new" title="Lipatan Hilbert (halaman belum tersedia)">lipatan Hilbert</a>), maka sampai pada gagasan tentang grup Lie berdimensi tak hingga. Dimungkinkan untuk mendefinisikan analogi dari banyak <a href="/w/index.php?title=Grup_tipe_Lie&action=edit&redlink=1" class="new" title="Grup tipe Lie (halaman belum tersedia)">grup Lie di atas bidang hingga</a>, dan memberikan sebagian besar contoh <a href="/w/index.php?title=Grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Grup sederhana hingga (halaman belum tersedia)">grup sederhana hingga</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Definisi_topologi">Definisi topologi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=5" title="Sunting bagian: Definisi topologi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=5" title="Sunting kode sumber bagian: Definisi topologi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grup Lie dapat didefinisikan sebagai (<a href="/w/index.php?title=Ruang_Hausdorff&action=edit&redlink=1" class="new" title="Ruang Hausdorff (halaman belum tersedia)">Hausdorff</a>) <a href="/wiki/Grup_topologi" title="Grup topologi">grup topologi</a> dimana elemen tersebut adalah identitas, terlihat seperti grup transformasi, tanpa referensi ke lipatan yang dibedakan.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Pertama, definisikan <b>grup Lie linear jauh</b> menjadi subgrup <i>G</i> dari grup linear umum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span> maka </p> <ol><li>untuk beberapa lingkungan <i>V</i> dari elemen identitas <i>e</i> dalam <i>G</i>, topologi <i>V</i> adalah topologi subruang <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span> dan <i>V</i> sebagai penutupan dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span>.</li> <li><i>G</i> memiliki <a href="/w/index.php?title=Himpunan_hitung&action=edit&redlink=1" class="new" title="Himpunan hitung (halaman belum tersedia)">hitung</a> komponen yang terhubung.</li></ol> <p>Misalnya, subgrup tertutup dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45c47c9aa8bae7df52f562932d18c8b0aa39a545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (n,\mathbb {C} )}"></span>; yaitu, matriks grup Lie memenuhi kondisi di atas. </p><p>Maka <i>grup Lie</i> didefinisikan sebagai grup topologi (1) secara lokal isomorfik dekat identitas ke grup Lie linear dan (2) memiliki banyak komponen yang terhubung. Menunjukkan definisi topologi ekuivalen dengan yang biasa bersifat teknis (dan pembaca pemula harus melewatkan yang berikut) tetapi dilakukan sebagai berikut: </p> <ol><li>Diberikan grup Lie <i> G </i> dalam arti berjenis biasa, <a href="/w/index.php?title=Korespondensi_grup_Lie%E2%80%93aljabar_Lie&action=edit&redlink=1" class="new" title="Korespondensi grup Lie–aljabar Lie (halaman belum tersedia)">korespondensi grup Lie–aljabar Lie</a> (atau versi <a href="/w/index.php?title=Teorema_ketiga_Lie&action=edit&redlink=1" class="new" title="Teorema ketiga Lie (halaman belum tersedia)">teorema ketiga Lie</a>) membentuk subgrup Lie terbenam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo>⊂</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae47832dc84b54cd683291957279fa229f383c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.803ex; height:3.009ex;" alt="{\displaystyle G'\subset \operatorname {GL} (n,\mathbb {C} )}"></span> maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>,</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G,G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c11c7db05254b6511cb5244edf569853eee37bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle G,G'}"></span> dibagikan aljabar Lie yang sama; dengan demikian, isomorfik secara lokal. Oleh karena itu, <i>G</i> memenuhi definisi topologi di atas.</li> <li>Maka <i>G</i> sebagai grup topologi yang merupakan grup Lie dalam pengertian topologis di atas dan grup Lie linear <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> lokal isomorfik ke <i>G</i>. Kemudian, dengan versi <a href="/w/index.php?title=Teorema_subgrup_tertutup&action=edit&redlink=1" class="new" title="Teorema subgrup tertutup (halaman belum tersedia)">teorema subgrup tertutup</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76634fad5818a777669a77cd8c86d1d816e4c402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle G'}"></span> adalah <a href="/w/index.php?title=Lipatan_analitik-riil&action=edit&redlink=1" class="new" title="Lipatan analitik-riil (halaman belum tersedia)">lipatan analitik-riil</a> dan isomorfisme lokal, <i>G</i> memperoleh struktur lipatan ganda dekat <a href="/wiki/Elemen_identitas" class="mw-redirect" title="Elemen identitas">elemen identitas</a>. Maka ditunjukkan hukum grup <i>G</i> diberikan deret pangkat formal;<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> jadi operasi grup adalah analitik-riil dan <i>G</i> adalah lipatan analitik-riil.</li></ol> <p>Definisi topologi sebagai dua grup Lie isomorfik sebagai grup topologi, maka isomorfik adalah grup Lie. Faktanya, prinsip umum bahwa untuk sebagian besar, <i>topologi grup Lie</i> dengan hukum grup menentukan geometri grup. </p> <div class="mw-heading mw-heading3"><h3 id="Contoh_pertama">Contoh pertama</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=6" title="Sunting bagian: Contoh pertama" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=6" title="Sunting kode sumber bagian: Contoh pertama"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Matriks_invers" class="mw-redirect" title="Matriks invers">Matriks</a> <a href="/wiki/Bilangan_riil" title="Bilangan riil">riil</a> 2 × 2 sebuah grup dalam perkalian, dilambangkan dengan <span class="nowrap"><a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">GL(2, <b>R</b>)</a></span> atau dengan GL<sub>2</sub>(<b>R</b>):</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\det A=ad-bc\neq 0\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>:</mo> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\det A=ad-bc\neq 0\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90be71d4fbdc0a440f3d0345868326abc3e647e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:53.021ex; height:6.176ex;" alt="{\displaystyle \operatorname {GL} (2,\mathbf {R} )=\left\{A={\begin{pmatrix}a&b\\c&d\end{pmatrix}}:\,\det A=ad-bc\neq 0\right\}.}"></span></dd></dl></dd></dl> <dl><dd>Ini disebut sebagai grup Lie riil empat dimensi <a href="/wiki/Ruang_kompak" title="Ruang kompak">non-kompak</a> adalah himpunan bagian dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span>. Grup ini <a href="/wiki/Ruang_terhubung" title="Ruang terhubung">menghubungkan</a> dua komponen diantara nilai positif dan negatif dari <a href="/wiki/Determinan" title="Determinan">determinan</a>.</dd></dl> <ul><li>Matriks <a href="/w/index.php?title=Rotasi_(matematika)&action=edit&redlink=1" class="new" title="Rotasi (matematika) (halaman belum tersedia)">rotasi</a> sebagai <a href="/wiki/Subgrup" title="Subgrup">subgrup</a> dari <span class="nowrap">GL(2, <b>R</b>)</span> yang dilambangkan dengan <span class="nowrap">SO(2, <b>R</b>)</span>. Ini disebut sebagai grup Lie dalam sendiri: khususnya, grup Lie menghubungkan kompak satu dimensi <a href="/w/index.php?title=Difeomorfik&action=edit&redlink=1" class="new" title="Difeomorfik (halaman belum tersedia)">difeomorfik</a> ke <a href="/wiki/Lingkaran" title="Lingkaran">lingkaran</a>. Menggunakan sudut rotasi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> sebagai parameter, grup ini dapat berupa <a href="/wiki/Persamaan_parametrik" title="Persamaan parametrik">parametrized</a> sebagai berikut:</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\,\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>SO</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>:</mo> <mspace width="thinmathspace" /> <mi>φ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\,\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c45036d7074bc34c66e4508239357a340c1c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.179ex; height:6.176ex;" alt="{\displaystyle \operatorname {SO} (2,\mathbf {R} )=\left\{{\begin{pmatrix}\cos \varphi &-\sin \varphi \\\sin \varphi &\cos \varphi \end{pmatrix}}:\,\varphi \in \mathbf {R} /2\pi \mathbf {Z} \right\}.}"></span></dd></dl></dd></dl> <dl><dd>Penjumlahan sudut sesuai dengan perkalian elemen <span class="nowrap">SO(2, <b>R</b>)</span>, dan mengambil sudut berlawanan sesuai dengan inversi. Jadi perkalian dan inversi adalah peta yang dapat dibedakan.</dd></dl> <ul><li><a href="/w/index.php?title=Grup_Affin&action=edit&redlink=1" class="new" title="Grup Affin (halaman belum tersedia)">Grup affin satu dimensi</a> adalah grup Lie matriks dua dimensi yang terdiri dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span> matriks segitiga atas dengan entri diagonal pertama positif dan entri diagonal kedua adalah 1. Jadi, grup tersebut terdiri dari matriks formulir</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821f7e2f4d606c47bd86a998a0456a5fce080333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.548ex; height:6.176ex;" alt="{\displaystyle A=\left({\begin{array}{cc}a&b\\0&1\end{array}}\right),\quad a>0,\,b\in \mathbb {R} .}"></span></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Bukan_contoh">Bukan contoh</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=7" title="Sunting bagian: Bukan contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=7" title="Sunting kode sumber bagian: Bukan contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Untuk contoh grup dengan elemen <a href="/w/index.php?title=Himpunan_tak_terhitung&action=edit&redlink=1" class="new" title="Himpunan tak terhitung (halaman belum tersedia)">tak terhitung</a> yang bukan grup Lie di bawah topologi tertentu. Grup diberikan oleh </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>a</mi> <mi>θ</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mi>ϕ</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>:</mo> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>,</mo> <mi>ϕ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9b797ab98232c2c00ab0ec3c98b881abd8538a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.997ex; height:6.509ex;" alt="{\displaystyle H=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi ia\theta }\end{matrix}}\right):\,\theta \in \mathbb {R} \right\}\subset \mathbb {T} ^{2}=\left\{\left({\begin{matrix}e^{2\pi i\theta }&0\\0&e^{2\pi i\phi }\end{matrix}}\right):\,\theta ,\phi \in \mathbb {R} \right\},}"></span></dd></dl> <p>dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52a8ffd71e08fa7d3229ee64ae27579a6ff5c8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.751ex; height:2.843ex;" alt="{\displaystyle a\in \mathbb {R} \setminus \mathbb {Q} }"></span> sebuah <a href="/wiki/Bilangan_irasional" title="Bilangan irasional">bilangan irasional</a> adalah subgrup dari <a href="/wiki/Torus" title="Torus">torus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e6c9160e26419c5bb5870e244b8fb50393051a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {T} ^{2}}"></span> yang bukan grup Lie diberikan oleh <a href="/w/index.php?title=Topologi_subruang&action=edit&redlink=1" class="new" title="Topologi subruang (halaman belum tersedia)">topologi subruang</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Jika mengambil <a href="/wiki/Lingkungan_(matematika)" title="Lingkungan (matematika)">lingkungan</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> dari sebuah titik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"></span> dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>: contoh, bagian dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> adalah terputus. Grup dengan rotasi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> di sekitar torus tanpa mencapai titik spiral sebelumnya dan dengan demikian sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e6c9160e26419c5bb5870e244b8fb50393051a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {T} ^{2}}"></span>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Irrational_line_on_a_torus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Irrational_line_on_a_torus.png/220px-Irrational_line_on_a_torus.png" decoding="async" width="220" height="222" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Irrational_line_on_a_torus.png/330px-Irrational_line_on_a_torus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/42/Irrational_line_on_a_torus.png 2x" data-file-width="360" data-file-height="364" /></a><figcaption>Sebagian dari grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {T} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e6c9160e26419c5bb5870e244b8fb50393051a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {T} ^{2}}"></span>. Lingkungan kecil dari elemen terputus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675a79e26028d91d97f4e2ce279c314b0f194c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.243ex; height:2.176ex;" alt="{\displaystyle h\in H}"></span> dalam himpunan bagian topologi dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span></figcaption></figure> <p>Grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> diberikan topologi yang berbeda, dimana jarak antara dua titik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1},h_{2}\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1},h_{2}\in H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81d21f1282f82ac1704d123a81f13cbce4330d01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.725ex; height:2.509ex;" alt="{\displaystyle h_{1},h_{2}\in H}"></span> didefinisikan sebagai panjang dari jalur terpendek <i>dalam grup</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> sebagai gabungan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}"></span> dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d2d1733d09f02f33694f72b6da94681021e0f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{2}}"></span>. Dalam topologi ini, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> diidentifikasi secara homeomorfis dengan garis riil untuk mengidentifikasi setiap elemen dengan bilangan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"></span> dalam definisi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. Dengan topologi ini, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> sebagai grup bilangan riil yang ditambahkan, oleh karena itu merupakan grup Lie. </p><p>Grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> adalah contoh gelanggang dari "<a href="#Subgrup_Lie">subgrup Lie</a>" dari grup Lie yang tidak tertutup. Lihat pembahasan subgrup Lie di bawah ini pada bagian tentang konsep dasar. </p> <div class="mw-heading mw-heading2"><h2 id="Lebih_banyak_contoh_dari_grup_Lie">Lebih banyak contoh dari grup Lie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=8" title="Sunting bagian: Lebih banyak contoh dari grup Lie" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=8" title="Sunting kode sumber bagian: Lebih banyak contoh dari grup Lie"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/w/index.php?title=Tabel_grup_Lie&action=edit&redlink=1" class="new" title="Tabel grup Lie (halaman belum tersedia)">Tabel grup Lie</a> dan <a href="/w/index.php?title=Daftar_grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Daftar grup Lie sederhana (halaman belum tersedia)">Daftar grup Lie sederhana</a></div> <p>Grup Lie terdapat di seluruh materi matematika dan fisika. <a href="/w/index.php?title=Grup_matriks&action=edit&redlink=1" class="new" title="Grup matriks (halaman belum tersedia)">Grup matriks</a> atau <a href="/w/index.php?title=Grup_aljabar&action=edit&redlink=1" class="new" title="Grup aljabar (halaman belum tersedia)">grup aljabar</a> adalah grup matriks, misalnya: <a href="/w/index.php?title=Grup_ortogonal&action=edit&redlink=1" class="new" title="Grup ortogonal (halaman belum tersedia)">ortogonal</a> dan <a href="/w/index.php?title=Grup_simplektis&action=edit&redlink=1" class="new" title="Grup simplektis (halaman belum tersedia)">grup simplektis</a>, dan ini memberikan sebagian besar yang umum contoh dari Lie. </p> <div class="mw-heading mw-heading3"><h3 id="Dimensi_satu_dan_dua">Dimensi satu dan dua</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=9" title="Sunting bagian: Dimensi satu dan dua" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=9" title="Sunting kode sumber bagian: Dimensi satu dan dua"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Salah satu grup Lie yang terhubung dengan dimensi satu adalah garis riil <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> dengan operasi grup menjadi penjumlahan dan <a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">grup lingkaran</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"></span> bilangan kompleks dengan nilai absolut satu dengan operasi grup menjadi perkalian. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"></span> grup dilambangkan sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62b00d74ee0cefb86cc052365625abff56d43e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.754ex; height:2.843ex;" alt="{\displaystyle U(1)}"></span> sebagai grup matriks uniter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}"></span>. </p><p>Dalam dua dimensi, jika membatasi hanya pada grup yang terhubung, maka diklasifikasikan oleh aljabar Lie. Ada (hingga isomorfisme) hanya dua aljabar Lie berdimensi dua. Grup Lie yang terhubung secara sederhana adalah <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> dengan operasi grup sebagai penjumlahan vektor dan grup affin dalam dimensi satu, dijelaskan di sub-bagian sebelumnya di bawah "contoh pertama". </p> <div class="mw-heading mw-heading3"><h3 id="Contoh_tambahan">Contoh tambahan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=10" title="Sunting bagian: Contoh tambahan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=10" title="Sunting kode sumber bagian: Contoh tambahan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Grup_uniter_khusus&action=edit&redlink=1" class="new" title="Grup uniter khusus (halaman belum tersedia)">Grup SU(2)</a> adalah grup matriks uniter <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"></span> dengan determinan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span>. Secara topologis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{SU}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>SU</mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{SU}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1850368e362fa0be265cbb948aca0ff64a83342" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.007ex; height:2.843ex;" alt="{\displaystyle {\text{SU}}(2)}"></span> adalah bola-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> oleh <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e57c690f890937838c10ba57853ff21bf30ec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{3}}"></span>; sebagai grup diidentifikasikan dengan grup unit <a href="/wiki/Kuaternion" title="Kuaternion">kuaternion</a>.</li> <li><a href="/w/index.php?title=Grup_Heisenberg&action=edit&redlink=1" class="new" title="Grup Heisenberg (halaman belum tersedia)">Grup Heisenberg</a> adalah grup dimensi <a href="/wiki/Grup_nilpoten" title="Grup nilpoten">nilpoten</a> menghubungkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"></span> yang memainkan peran kunci dalam <a href="/wiki/Mekanika_kuantum" title="Mekanika kuantum">mekanika kuantum</a>.</li> <li><a href="/w/index.php?title=Gru0_Lorentz&action=edit&redlink=1" class="new" title="Gru0 Lorentz (halaman belum tersedia)">Gru0 Lorentz</a> adalah grup Lie 6 dimensi dari <a href="/wiki/Isometri" title="Isometri">isometri</a> dari <a href="/wiki/Ruang_Minkowski" title="Ruang Minkowski">ruang Minkowski</a>.</li> <li><a href="/w/index.php?title=Grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Grup Poincaré (halaman belum tersedia)">Grup Poincaré</a> adalah grup Lie 10 dimensi dari isometri <a href="/w/index.php?title=Transformasi_affin&action=edit&redlink=1" class="new" title="Transformasi affin (halaman belum tersedia)">affin</a> dari ruang Minkowski.</li> <li><a href="/w/index.php?title=Grup_Lie_eksepsional&action=edit&redlink=1" class="new" title="Grup Lie eksepsional (halaman belum tersedia)">Grup Lie eksepsional</a> tipe <a href="/w/index.php?title=G2_(matematika)&action=edit&redlink=1" class="new" title="G2 (matematika) (halaman belum tersedia)"><i>G</i><sub>2</sub></a>, <a href="/w/index.php?title=F4_(matematika)&action=edit&redlink=1" class="new" title="F4 (matematika) (halaman belum tersedia)"><i>F</i><sub>4</sub></a>, <a href="/w/index.php?title=E6_(matematika)&action=edit&redlink=1" class="new" title="E6 (matematika) (halaman belum tersedia)"><i>E</i><sub>6</sub></a>, <a href="/w/index.php?title=E7_(matematika)&action=edit&redlink=1" class="new" title="E7 (matematika) (halaman belum tersedia)"><i>E</i><sub>7</sub></a>, <a href="/w/index.php?title=E8_(matematika)&action=edit&redlink=1" class="new" title="E8 (matematika) (halaman belum tersedia)"><i>E</i><sub>8</sub></a> memiliki dimensi 14, 52, 78, 133, dan 248. Dengan deret A-B-C-D <a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">grup Lie sederhana</a>, grup eksepsional melengkapi daftar grup Lie sederhana.</li> <li><a href="/w/index.php?title=Grup_simplektik&action=edit&redlink=1" class="new" title="Grup simplektik (halaman belum tersedia)">Grup simplektik</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Sp}}(2n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Sp</mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Sp}}(2n,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f654ae45628075171352047380bbfe8a5493bdf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.664ex; height:2.843ex;" alt="{\displaystyle {\text{Sp}}(2n,\mathbb {R} )}"></span> terdiri dari semua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n\times 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>×</mo> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n\times 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f264bcf2cf920cb907b8c9a50120da5d34a27a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.955ex; height:2.176ex;" alt="{\displaystyle 2n\times 2n}"></span> matriks mempererat <i><a href="/w/index.php?title=Bentuk_simplektis&action=edit&redlink=1" class="new" title="Bentuk simplektis (halaman belum tersedia)">bentuk simplektis</a></i> dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47460f1a92774729807be11cf62b9178b5771b4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.719ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2n}}"></span>. Ini disebut sebagai grup dimensi Lie yang menghubungkan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2n^{2}+n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n^{2}+n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f666e0b56b0e831b0a1c3d2c07df2b69f2016682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.847ex; height:2.843ex;" alt="{\displaystyle 2n^{2}+n}"></span>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Konstruksi">Konstruksi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=11" title="Sunting bagian: Konstruksi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=11" title="Sunting kode sumber bagian: Konstruksi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ada beberapa cara standar untuk membentuk grup Lie yang baru dari lama: </p> <ul><li>Produk dari dua grup Lie adalah grup Lie.</li> <li>Setiap subgrup <a href="/w/index.php?title=Himpunan_tertutup&action=edit&redlink=1" class="new" title="Himpunan tertutup (halaman belum tersedia)">topologi tertutup</a> dari grup Lie adalah grup Lie. Ini dikenal sebagai <a href="/w/index.php?title=Teorema_subgrup_tertutup&action=edit&redlink=1" class="new" title="Teorema subgrup tertutup (halaman belum tersedia)">Teorema subgrup tertutup</a> atau <b>teorema Cartan</b>.</li> <li>Hasil bagi dari grup Lie oleh subgrup normal tertutup adalah grup Lie.</li> <li><a href="/w/index.php?title=Sampul_universal&action=edit&redlink=1" class="new" title="Sampul universal (halaman belum tersedia)">Sampul universal</a> dari grup Lie yang terhubung adalah grup Lie. Misalnya, grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> adalah sampul universal grup lingkaran <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60796c8d0c03cf575637d3202463b214d9635880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}"></span>. Faktanya, setiap simpul dari lipatan yang dapat dibedakan juga merupakan lipatan yang dapat dibedakan, tetapi dengan menentukan sampul <i>universal</i> untuk struktur grup (kompatibel dengan struktur lainnya).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Pengertian_terkait">Pengertian terkait</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=12" title="Sunting bagian: Pengertian terkait" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=12" title="Sunting kode sumber bagian: Pengertian terkait"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Beberapa contoh grup yang <i>bukan</i> grup Lie (kecuali dalam pengertian solvabel bahwa setiap grup banyak dapat dilihat sebagai grup Lie 0 dimensi, dengan <a href="/w/index.php?title=Topologi_diskrit&action=edit&redlink=1" class="new" title="Topologi diskrit (halaman belum tersedia)">topologi diskrit</a>), adalah: </p> <ul><li>Gugus berdimensi tak hingga merupakan grup aditif <a href="/wiki/Ruang_vektor" title="Ruang vektor">ruang vektor</a> riil berdimensi tak hingga, atau ruang fungsi halus dari lipatan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> ke grup Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(X,G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\infty }(X,G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c8b240910093f8e5e56d8977764d60acb17b41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.323ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(X,G)}"></span>. Ini bukan grup Lie karena bukan lipatan "berdimensi-hingga".</li> <li>Beberapa <a href="/w/index.php?title=Grup_total_putusan&action=edit&redlink=1" class="new" title="Grup total putusan (halaman belum tersedia)">grup total putusan</a> merupakan <a href="/wiki/Grup_Galois" title="Grup Galois">grup Galois</a> dengan ekstensi tak hingga bidang, atau grup aditif dari bilangan <i>p</i>-adik. Ini bukan grup Lie karena ruang dasarnya bukan lipatan riil. Beberapa dari grup ini adalah "grup Lie <i>p</i>-adik". Secara umum, l grup topologi yang memiliki kesamaan <a href="/w/index.php?title=Sifat_lokal&action=edit&redlink=1" class="new" title="Sifat lokal (halaman belum tersedia)">sifat lokal</a> <b>R</b><sup><i>n</i></sup> untuk beberapa bilangan bulat positif <i>n</i> dapat berupa grup Lie (tentu harus memiliki struktur yang dibedakan).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Konsep_dasar">Konsep dasar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=13" title="Sunting bagian: Konsep dasar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=13" title="Sunting kode sumber bagian: Konsep dasar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Peta_eksponensial">Peta eksponensial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=14" title="Sunting bagian: Peta eksponensial" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=14" title="Sunting kode sumber bagian: Peta eksponensial"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Peta_eksponensial_(teori_Lie)&action=edit&redlink=1" class="new" title="Peta eksponensial (teori Lie) (halaman belum tersedia)">Peta eksponensial (teori Lie)</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/w/index.php?title=Turunan_dari_peta_eksponensial&action=edit&redlink=1" class="new" title="Turunan dari peta eksponensial (halaman belum tersedia)">Turunan dari peta eksponensial</a> dan <a href="/w/index.php?title=Koordinat_normal&action=edit&redlink=1" class="new" title="Koordinat normal (halaman belum tersedia)">koordinat normal</a></div> <p><a href="/w/index.php?title=Peta_eksponensial_(teori_Lie)&action=edit&redlink=1" class="new" title="Peta eksponensial (teori Lie) (halaman belum tersedia)">Peta eksponensial</a> untuk aljabar Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n;\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n;\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/834b969e8eeb4f35c41c2ffb58d6f40a5c8a7473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.358ex; height:2.843ex;" alt="{\displaystyle M(n;\mathbb {C} )}"></span> dari <a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">grup linear umum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(n;\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(n;\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9776690f92025c8e4d2fecfec95fd8ae79428804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.326ex; height:2.843ex;" alt="{\displaystyle GL(n;\mathbb {C} )}"></span> ke <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(n;\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(n;\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9776690f92025c8e4d2fecfec95fd8ae79428804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.326ex; height:2.843ex;" alt="{\displaystyle GL(n;\mathbb {C} )}"></span> ditentukan dengan <a href="/w/index.php?title=Matriks_eksponensial&action=edit&redlink=1" class="new" title="Matriks eksponensial (halaman belum tersedia)">matriks eksponensial</a> yang diberikan oleh deret pangkat biasa untuk matriks <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>X</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mn>3</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09e58dbc8c40e12db1a0091ac9008b54dc9d5a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.442ex; height:5.843ex;" alt="{\displaystyle \exp(X)=1+X+{\frac {X^{2}}{2!}}+{\frac {X^{3}}{3!}}+\cdots }"></span></dd></dl> <p>Jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> adalah subgrup tertutup dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(n;\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(n;\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9776690f92025c8e4d2fecfec95fd8ae79428804" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.326ex; height:2.843ex;" alt="{\displaystyle GL(n;\mathbb {C} )}"></span>, maka peta eksponensial mengambil aljabar Lie dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> menjadi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>; dengan demikian, memiliki peta eksponensial untuk semua grup matriks. Setiap elemen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> yang hampir dekat dengan identitas adalah eksponensial matriks dalam aljabar Lie.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Definisi di atas mudah digunakan, tetapi tidak ditentukan untuk grup Lie yang bukan grup matriks, dan tidak jelas bahwa peta eksponensial grup Lie tidak bergantung pada wakilannya. Kita dapat menyelesaikan kedua masalah tersebut menggunakan definisi yang abstrak dari peta eksponensial yang berfungsi untuk semua grup Lie, sebagai berikut. </p><p>Untuk setiap vektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> dalam aljabar Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> yaitu ruang bersinggungan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> pada identitas, yang membuktikan bahwa subgrup satu parameter unik <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c:\mathbb {R} \rightarrow G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c:\mathbb {R} \rightarrow G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813c4cab3631fb425107e9e27a154d39e08b5140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.063ex; height:2.176ex;" alt="{\displaystyle c:\mathbb {R} \rightarrow G}"></span> dirumuskan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c'(0)=X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c'(0)=X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d141014eeee76f6b152796ab9aae26614606026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.742ex; height:3.009ex;" alt="{\displaystyle c'(0)=X}"></span>. Bahwa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> adalah subgrup satu parameter berarti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> adalah peta mulus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> dan untuk semua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(s+t)=c(s)c(t)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(s+t)=c(s)c(t)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a250d6a659170920841ad7e1d7999432f7f8d7bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.828ex; height:2.843ex;" alt="{\displaystyle c(s+t)=c(s)c(t)\ }"></span></dd></dl> <p>Operasi di sisi kanan adalah perkalian grup dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Kesamaan formal rumus ini dengan yang valid untuk <a href="/wiki/Fungsi_eksponensial" title="Fungsi eksponensial">fungsi eksponensial</a> membenarkan definisi tersebut </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)=c(1).\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(X)=c(1).\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56af124e5c4ec91da6a749d5bf3849c6f7da036" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.033ex; height:2.843ex;" alt="{\displaystyle \exp(X)=c(1).\ }"></span></dd></dl> <p>Ini disebut <b>peta eksponensial</b>, dan memetakan aljabar Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span> dalam grup Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Ini memberikan <a href="/w/index.php?title=Diffeomorfisme&action=edit&redlink=1" class="new" title="Diffeomorfisme (halaman belum tersedia)">diffeomorfisme</a> antara <a href="/w/index.php?title=Lingkungan_(topologi)&action=edit&redlink=1" class="new" title="Lingkungan (topologi) (halaman belum tersedia)">lingkungan</a> dari 0 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span> dan lingkungan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> dalam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Peta eksponensial ini merupakan generalisasi dari fungsi eksponensial untuk bilangan riil, maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> adalah aljabar Lie dari kelompok Lie <a href="/w/index.php?title=Bilangan_riil_positif&action=edit&redlink=1" class="new" title="Bilangan riil positif (halaman belum tersedia)">bilangan riil positif</a> dengan perkalian, untuk bilangan kompleks, maka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> adalah aljabar Lie dari grup Lie dari bilangan kompleks bukan nol dengan perkalian) dan untuk <a href="/wiki/Matriks_(matematika)" title="Matriks (matematika)">matriks</a> (karena <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57e3220831cc96b462d9e4a480ec0360888c897" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.358ex; height:2.843ex;" alt="{\displaystyle M(n,\mathbb {R} )}"></span> dengan komutator biasa adalah aljabar Lie dari grup Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(n,\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f7197960fac26cadfe027d3045154b9972f8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.326ex; height:2.843ex;" alt="{\displaystyle GL(n,\mathbb {R} )}"></span> dari semua matriks invers). </p><p>Karena peta eksponensial bersifat konjektur di beberapa lingkungan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> adalah hal umum untuk elemen aljabar Lie <b>infinitesimal generator</b> dari grup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. Subgrup <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> adalah komponen identitas <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p><p>Peta eksponensial dan aljabar Lie menentukan <i>struktur grup lokal</i> dari setiap grup Lie yang terhubung, karena <a href="/w/index.php?title=Rumus_Baker%E2%80%93Campbell%E2%80%93Hausdorff&action=edit&redlink=1" class="new" title="Rumus Baker–Campbell–Hausdorff (halaman belum tersedia)">rumus Baker–Campbell–Hausdorff</a>: lingkungan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> dari elemen nol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span> yang dirumuskan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X,Y\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12df4fb6ff6913669ce621b6d41645366d5f7e27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.41ex; height:2.509ex;" alt="{\displaystyle X,Y\in U}"></span>, maka </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">[</mo> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>X</mi> <mo stretchy="false">]</mo> <mo>−</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb833ea1dab7d320bb56e8c0f98a29aa138c05d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:79.401ex; height:3.509ex;" alt="{\displaystyle \exp(X)\,\exp(Y)=\exp \left(X+Y+{\tfrac {1}{2}}[X,Y]+{\tfrac {1}{12}}[\,[X,Y],Y]-{\tfrac {1}{12}}[\,[X,Y],X]-\cdots \right),}"></span></dd></dl> <p>dimana istilah yang dihilangkan diketahui dan melibatkan kurung Lie dari empat elemen atau lebih. Jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> komutator, rumus tersebut direduksi menjadi hukum eksponensial yang dikenal sebagai <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539f461f3d315c25c8d7a5c9913741562c30d151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.918ex; height:2.843ex;" alt="{\displaystyle \exp(X)\exp(Y)=\exp(X+Y)}"></span> </p><p>Peta eksponensial menghubungkan homomorfisme grup Lie. Artinya, jika <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :G\to H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :G\to H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7a655e5848542428a585814efb99648f189d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.827ex; height:2.509ex;" alt="{\displaystyle \phi :G\to H}"></span> adalah homomorfisme grup Lie dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d339e710b6608192b94d0e52ff010e9d677640d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.374ex; height:2.509ex;" alt="{\displaystyle \phi _{*}:{\mathfrak {g}}\to {\mathfrak {h}}}"></span> peta induksi aljabar Lie yang tepat, maka untuk semua <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0f2fbc85e0c0a5747dc189789423f21695a43f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle x\in {\mathfrak {g}}}"></span> yaitu </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (\exp(x))=\exp(\phi _{*}(x)).\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\exp(x))=\exp(\phi _{*}(x)).\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b940af3153a224f3516dac19651ddc6a41bdd8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.959ex; height:2.843ex;" alt="{\displaystyle \phi (\exp(x))=\exp(\phi _{*}(x)).\,}"></span></dd></dl> <p>Dengan kata lain, diagram berikut <a href="/wiki/Diagram_komutatif" title="Diagram komutatif">komutatif</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>Catatan 1<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/Berkas:ExponentialMap-01.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/06/ExponentialMap-01.png" decoding="async" width="155" height="136" class="mw-file-element" data-file-width="155" data-file-height="136" /></a><figcaption></figcaption></figure> <p>Singkatnya, <i>exp</i> adalah <a href="/w/index.php?title=Transformasi_alami&action=edit&redlink=1" class="new" title="Transformasi alami (halaman belum tersedia)">transformasi alami</a> dari functor Lie ke identitas funktor pada kategori grup Lie. </p><p>Peta eksponensial dari aljabar Lie ke grup Lie tidak selalu <a href="/w/index.php?title=Fungsi_ekspresif&action=edit&redlink=1" class="new" title="Fungsi ekspresif (halaman belum tersedia)">ekspresif</a>, bahkan jika grup tersebut terhubung yang memetakan ke grup Lie untuk grup terhubung yang kompak atau nilpoten. </p> <div class="mw-heading mw-heading3"><h3 id="Subgrup_Lie">Subgrup Lie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=15" title="Sunting bagian: Subgrup Lie" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=15" title="Sunting kode sumber bagian: Subgrup Lie"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Subgrup Lie</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> dari grup Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> adalah grup Lie <a href="/wiki/Himpunan_bagian" title="Himpunan bagian">himpunan bagian</a> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> dan <a href="/w/index.php?title=Peta_inklusi&action=edit&redlink=1" class="new" title="Peta inklusi (halaman belum tersedia)">peta inklusi</a> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> ke <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> yang merupakan <a href="/w/index.php?title=Injektif&action=edit&redlink=1" class="new" title="Injektif (halaman belum tersedia)">injektif</a> <a href="/w/index.php?title=Perendaman_(matematika)&action=edit&redlink=1" class="new" title="Perendaman (matematika) (halaman belum tersedia)">pencelupan</a> dan <a href="/wiki/Homomorfisme_grup" class="mw-redirect" title="Homomorfisme grup">homomorfisme grup</a>. Menurut <a href="/w/index.php?title=Teorema_subgrup_tertutup&action=edit&redlink=1" class="new" title="Teorema subgrup tertutup (halaman belum tersedia)">teorema Cartan</a>, <a href="/wiki/Subgrup" title="Subgrup">subgrup</a> tertutup dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> mengetahui struktur halus unik yang menjadikannya sebuah subgrup <a href="/w/index.php?title=Penyematan&action=edit&redlink=1" class="new" title="Penyematan (halaman belum tersedia)">tancapan</a> Lie dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, yaitu sebuah subgrup Lie sedemikian rupa sehingga peta inklusi adalah penyematan mulus. </p><p>Banyak contoh subgrup non-tertutup; misalnya mengambil <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> sebagai torus berdimensi 2 atau lebih besar, dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> sebagai <a href="/w/index.php?title=Subgrup_satu_parameter&action=edit&redlink=1" class="new" title="Subgrup satu parameter (halaman belum tersedia)">subgrup satu parameter</a> dari <i>lerengan irasional</i>, yaitu salah satu dalam <i>G</i>. Maka grup Lie <a href="/wiki/Homomorfisme" title="Homomorfisme">homomorfisme</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :\mathbb {R} \to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :\mathbb {R} \to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2befb1d96d8436a0631380dc3f0ada90d36f4c94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.576ex; height:2.676ex;" alt="{\displaystyle \varphi :\mathbb {R} \to G}"></span> dengan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {im} (\varphi )=H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">m</mi> </mrow> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {im} (\varphi )=H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb71bbd8ae1dbb483bc17f7c9b7ee6119f5d7e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.074ex; height:2.843ex;" alt="{\displaystyle \mathrm {im} (\varphi )=H}"></span>. <a href="/w/index.php?title=Penutupan_(topologi)&action=edit&redlink=1" class="new" title="Penutupan (topologi) (halaman belum tersedia)">penutupan</a> dari <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> sebagai sub-torus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p><p><a href="/w/index.php?title=Peta_eksponensial_(teori_Lie)&action=edit&redlink=1" class="new" title="Peta eksponensial (teori Lie) (halaman belum tersedia)">peta eksponensial</a> menghasilkan <a href="/w/index.php?title=Korespondensi_aljabar_Lie%E2%80%93grup_Lie&action=edit&redlink=1" class="new" title="Korespondensi aljabar Lie–grup Lie (halaman belum tersedia)">korespondensi satu-ke-satu</a> antara subgrup Lie terhubung dari grup Lie yang terhubung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> dan subaljabar dari aljabar Lie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Biasanya, subgrup yang sesuai dengan subaljabar bukanlah subgrup tertutup. Tidak ada kriteria yang didasarkan pada struktur <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> untuk menentukan subaljabar, dimana yang sesuai dengan subgrup tertutup. </p> <div class="mw-heading mw-heading2"><h2 id="Wakilan">Wakilan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=16" title="Sunting bagian: Wakilan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=16" title="Sunting kode sumber bagian: Wakilan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel utama: <a href="/w/index.php?title=Wakilan_dari_grup_Lie&action=edit&redlink=1" class="new" title="Wakilan dari grup Lie (halaman belum tersedia)">Wakilan dari grup Lie</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Lihat pula: <a href="/w/index.php?title=Grup_kompak&action=edit&redlink=1" class="new" title="Grup kompak (halaman belum tersedia)">Grup kompak § Teori wakilan dari grup Lie kompak terhubung</a>, dan <a href="/w/index.php?title=Wakilan_aljabar_Lie&action=edit&redlink=1" class="new" title="Wakilan aljabar Lie (halaman belum tersedia)">Wakilan aljabar Lie</a></div> <p>Salah satu aspek penting dari studi grup Lie adalah wakilan, yaitu cara bertindak (secara linear) pada ruang vektor. Dalam fisika, grup Lie sering kali menyandikan kesimetrian sistem fisik. Cara menggunakan simetri ini untuk membantu menganalisis sistem sering kali melalui teori wakilan. Pertimbangkan, misalnya, <a href="/wiki/Persamaan_Schr%C3%B6dinger" title="Persamaan Schrödinger">persamaan Schrödinger</a> yang tidak bergantung waktu dalam mekanika kuantum, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}\psi =E\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi =E\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b9995521be4f5e023d9cec5087f07f99fb4516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.964ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}\psi =E\psi }"></span>. Asumsikan sistem yang dimaksud <a href="/w/index.php?title=Grup_rotasi_SO(3)&action=edit&redlink=1" class="new" title="Grup rotasi SO(3) (halaman belum tersedia)">grup rotasi SO(3)</a> sebagai simetri, artinya operasi Hamiltonian <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb06de5217295d7fbdbf68fb9c5309a513fc99e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.843ex;" alt="{\displaystyle {\hat {H}}}"></span> komutatif dengan aksi SO(3) pada fungsi gelombang <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span>. Salah satu contoh penting dari sistem hal itu adalah <a href="/wiki/Atom_hidrogen" title="Atom hidrogen">atom hidrogen</a>. Asumsi tersebut tidak berarti bahwa solusi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> adalah fungsi invarian secara rotasi. Sebaliknya, hal itu berarti bahwa <i>ruang</i> dari solusi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {H}}\psi =E\psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>H</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mi>ψ</mi> <mo>=</mo> <mi>E</mi> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {H}}\psi =E\psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b9995521be4f5e023d9cec5087f07f99fb4516" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.964ex; height:3.176ex;" alt="{\displaystyle {\hat {H}}\psi =E\psi }"></span> adalah invarian dalam rotasi (untuk setiap nilai tetap <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>). Ruang ini, merupakan wakilan dari SO(3). Wakilan ini telah <a href="/w/index.php?title=Wakilan_grup_Lie&action=edit&redlink=1" class="new" title="Wakilan grup Lie (halaman belum tersedia)">diklasifikasikan</a> dan mengarah ke penyederhanaan <a href="/wiki/Atom_bakhidrogen" title="Atom bakhidrogen">penyederhanaan masalah</a>, pada dasarnya mengubah persamaan diferensial parsial tiga dimensi menjadi persamaan diferensial biasa satu dimensi. </p><p>Kasus grup Lie kompak terhubung <i>K</i> (termasuk kasus SO(3) yang baru saja disebutkan) sangat mudah ditangani.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Dalam hal ini, setiap wakilan berdimensi-hingga dari <i>K</i> terurai sebagai jumlah langsung dari wakilan yang tidak direduksi. Wakilan yang tidak direduksi, pada gilirannya, diklasifikasikan oleh <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>. <a href="/w/index.php?title=Grup_kompak&action=edit&redlink=1" class="new" title="Grup kompak (halaman belum tersedia)">Klasifikasi</a> adalah dalam istilah "bobot tertinggi" dari representasi. Klasifikasi ini terkait erat dengan <a href="/w/index.php?title=Wakilan_aljabar_Lie&action=edit&redlink=1" class="new" title="Wakilan aljabar Lie (halaman belum tersedia)">klasifikasi wakilan dari aljabar Lie semisederhana</a>. </p><p>Dengan mempelajari wakilan satuan (secara umum berdimensi-tak-hingga) dari suatu grup Lie yang berubah-ubah (tidak kompak). Misalnya, untuk memberikan deskripsi eksplisit yang relatif sederhana tentang <a href="/w/index.php?title=Teori_wakilan_SL2(R)&action=edit&redlink=1" class="new" title="Teori wakilan SL2(R) (halaman belum tersedia)">wakilan dari grup SL(2,R)</a> dan <a href="/w/index.php?title=Klasifikasi_Wigner%27&action=edit&redlink=1" class="new" title="Klasifikasi Wigner' (halaman belum tersedia)">wakilan dari grup Poincaré</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Sejarah_awal">Sejarah awal</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=17" title="Sunting bagian: Sejarah awal" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=17" title="Sunting kode sumber bagian: Sejarah awal"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Menurut sumber paling otoritatif pada sejarah awal grup Lie (Hawkins, hal. 1), <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> menganggap musim dingin tahun 1873–1874 sebagai tanggal lahir teorinya tentang grup kontinu. Namun, Hawkins menyatakan bahwa "aktivitas penelitian Lie yang luar biasa selama periode empat tahun dari musim gugur 1869 hingga musim gugur 1873" yang mengarah pada penciptaan teori (<i>ibid</i>). Beberapa ide awal Lie dikembangkan dalam kolaborasi erat dengan <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>. Lie bertemu dengan Klein setiap hari dari Oktober 1869 hingga 1872 di Berlin dari akhir Oktober 1869 hingga akhir Februari 1870, dan di Paris, Göttingen dan Erlangen dalam dua tahun berikutnya (<i>ibid</i>, hal. 2). Lie menyatakan bahwa semua hasil utama diperoleh pada tahun 1884. Tetapi selama tahun 1870-an semua makalahnya (kecuali catatan pertama) diterbitkan di jurnal Norwegia yang menghambat pengakuan atas karya tersebut di seluruh Eropa (<i>ibid</i>, hal 76). Pada tahun 1884, matematikawan muda asal Jerman, <a href="/wiki/Friedrich_Engel_(matematikawan)" title="Friedrich Engel (matematikawan)">Friedrich Engel</a>, datang untuk bekerja dengan Lie pada risalah sistematis untuk mengekspos teorinya tentang grup kontinu. Dari upaya ini dihasilkan tiga jilid Theorie der Transformationsgruppen, diterbitkan pada tahun 1888, 1890, dan 1893. Istilah <i>groupes de Lie</i> pertama kali muncul dalam bahasa Prancis pada tahun 1893 dalam tesis murid Lie, Arthur Tresse.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>Ide Lie tidak terpisah dari matematika lainnya. Faktanya, ketertarikannya pada geometri persamaan diferensial pertama kali dimotivasi oleh karya <a href="/w/index.php?title=Carl_Gustav_Jacobi&action=edit&redlink=1" class="new" title="Carl Gustav Jacobi (halaman belum tersedia)">Carl Gustav Jacobi</a>, pada teori <a href="/wiki/Persamaan_diferensial_parsial" title="Persamaan diferensial parsial">persamaan diferensial parsial</a> orde pertama dan pada persamaan <a href="/wiki/Mekanika_klasik" title="Mekanika klasik">mekanika klasik</a>. Banyak dari karya Jacobi diterbitkan secara anumerta pada tahun 1860-an, membangkitkan minat yang sangat besar di Prancis dan Jerman (Hawkins, hal.43). <i>Idée fixe</i> Lie adalah pengembangan teori kesimetrian persamaan diferensial yang diselesaikan oleh <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> untuk persamaan aljabar: yaitu, untuk mengklasifikasikannya dalam teori grup. Lie dan matematikawan lainnya menunjukkan persamaan yang paling penting untuk <a href="/w/index.php?title=Fungsi_khusus&action=edit&redlink=1" class="new" title="Fungsi khusus (halaman belum tersedia)">fungsi khusus</a> dan <a href="/w/index.php?title=Polinomial_ortogonal&action=edit&redlink=1" class="new" title="Polinomial ortogonal (halaman belum tersedia)">polinomial ortogonal</a> cenderung muncul dari kesimetrian teoretis grup. Dalam karya awal Lie, idenya adalah untuk membangun teori <i>grup kontinu</i>, untuk melengkapi teori <a href="/w/index.php?title=Kelompok_diskrit&action=edit&redlink=1" class="new" title="Kelompok diskrit (halaman belum tersedia)">kelompok diskrit</a> yang telah dikembangkan dalam teori <a href="/wiki/Bentuk_modular" title="Bentuk modular">bentuk modular</a>, di tangan <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> dan <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a>. Aplikasi awal yang ada dalam pikiran Lie adalah teori <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a>. Pada model <a href="/wiki/Teori_Galois" title="Teori Galois">teori Galois</a> dan <a href="/w/index.php?title=Persamaan_polinomial&action=edit&redlink=1" class="new" title="Persamaan polinomial (halaman belum tersedia)">persamaan polinomial</a>, konsep penggeraknya adalah teori yang mampu menyatukan, dengan mempelajari <a href="/wiki/Simetri" title="Simetri">simetri</a>, seluruh luas <a href="/wiki/Persamaan_diferensial_biasa" title="Persamaan diferensial biasa">persamaan diferensial biasa</a>. Namun, harapan bahwa Teori Kebohongan akan menyatukan seluruh bidang persamaan diferensial biasa tidak terpenuhi. Metode simetri untuk ODE terus dipelajari, namun tidak mendominasi materi. Ada <a href="/w/index.php?title=Teori_Galois_diferensial&action=edit&redlink=1" class="new" title="Teori Galois diferensial (halaman belum tersedia)">teori Galois diferensial</a>, tetapi dikembangkan oleh orang lain, seperti Picard dan Vessiot, dan ini memberikan teori <a href="/w/index.php?title=Kuadratur_(matematika)&action=edit&redlink=1" class="new" title="Kuadratur (matematika) (halaman belum tersedia)">kuadratur</a>, <a href="/w/index.php?title=Integral_tak_hingga&action=edit&redlink=1" class="new" title="Integral tak hingga (halaman belum tersedia)">integral tak hingga</a>. </p><p>Dorongan tambahan untuk mempertimbangkan kelompok berkelanjutan berasal dari gagasan <a href="/wiki/Bernhard_Riemann" class="mw-redirect" title="Bernhard Riemann">Bernhard Riemann</a>, pada dasar-dasar geometri, dan pengembangan lebih lanjut mereka di tangan Klein. Jadi tiga tema utama dalam matematika abad ke-19 digabungkan oleh Lie dalam menciptakan teori barunya: ide simetri, seperti yang dicontohkan oleh Galois melalui pengertian aljabar dari <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a>; teori geometri dan solusi eksplisit dari <a href="/wiki/Persamaan_diferensial" title="Persamaan diferensial">persamaan diferensial</a> mekanika, dikerjakan oleh <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a> dan Jacobi; dan pemahaman baru tentang <a href="/wiki/Geometri" title="Geometri">geometri</a> yang muncul dalam karya <a href="/w/index.php?title=Julius_Pl%C3%BCcker&action=edit&redlink=1" class="new" title="Julius Plücker (halaman belum tersedia)">Plücker</a>, <a href="/wiki/August_Ferdinand_M%C3%B6bius" title="August Ferdinand Möbius">Möbius</a>, <a href="/w/index.php?title=Grassmann&action=edit&redlink=1" class="new" title="Grassmann (halaman belum tersedia)">Grassmann</a> dan lainnya, dan berpuncak pada visi revolusioner Riemann tentang subjek tersebut. </p><p>Meskipun saat ini Sophus Lie diakui sebagai pencipta teori kelompok berkelanjutan, langkah besar dalam pengembangan teori struktur mereka, yang memiliki pengaruh besar pada perkembangan matematika selanjutnya, dibuat oleh <a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a>, yang pada tahun 1888 menerbitkan makalah pertama dalam seri berjudul <i>Die Zusammensetzung der stetigen endlichen Transformationsgruppen</i> (<i>Komposisi grup transformasi hingga kontinu</i>) (Hawkins, hlm. 100). Pekerjaan Pembunuhan, kemudian disempurnakan dan digeneralisasikan oleh <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>, mengarah ke klasifikasi <a href="/w/index.php?title=Aljabar_Lie_setengah_sederhana&action=edit&redlink=1" class="new" title="Aljabar Lie setengah sederhana (halaman belum tersedia)">aljabar Lie setengah sederhana</a>, Teori Cartan tentang <a href="/w/index.php?title=Ruang_simetris_Riemannian&action=edit&redlink=1" class="new" title="Ruang simetris Riemannian (halaman belum tersedia)">ruang simetris</a>, dan deskripsi <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> tentang <a href="/w/index.php?title=Representasi_kelompok&action=edit&redlink=1" class="new" title="Representasi kelompok (halaman belum tersedia)">representasi</a> dari grup Lie yang kompak dan setengah sederhana. </p><p>Pada tahun 1900 <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> menantang ahli teori Lie dengan <a href="/w/index.php?title=Masalah_kelima_Hilbert&action=edit&redlink=1" class="new" title="Masalah kelima Hilbert (halaman belum tersedia)">Masalah Kelima</a> yang dipresentasikan pada <a href="/w/index.php?title=Kongres_Internasional_Ahli_Matematika&action=edit&redlink=1" class="new" title="Kongres Internasional Ahli Matematika (halaman belum tersedia)">Kongres Internasional Ahli Matematika</a> di Paris. </p><p>Weyl membawa periode awal perkembangan teori kelompok Lie membuahkan hasil, karena tidak hanya dia mengklasifikasikan representasi tak tersederhanakan dari kelompok Lie semisimple dan menghubungkan teori grup dengan mekanika kuantum, tetapi dia juga menempatkan teori Lie itu sendiri pada pijakan yang lebih kokoh dengan secara jelas menyatakan perbedaan antara <i> grup sangat kecil </i> Lie (yaitu, Lie algebras) dan grup Lie yang sesuai, dan mulai menyelidiki topologi grup Lie.<sup id="cite_ref-FOOTNOTEBorel2001_12-0" class="reference"><a href="#cite_note-FOOTNOTEBorel2001-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Teori kelompok Lie secara sistematis dikerjakan ulang dalam bahasa matematika modern dalam sebuah monograf oleh <a href="/w/index.php?title=Claude_Chevalley&action=edit&redlink=1" class="new" title="Claude Chevalley (halaman belum tersedia)">Claude Chevalley</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Konsep_grup_Lie,_dan_kemungkinan_klasifikasi"><span id="Konsep_grup_Lie.2C_dan_kemungkinan_klasifikasi"></span>Konsep grup Lie, dan kemungkinan klasifikasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=18" title="Sunting bagian: Konsep grup Lie, dan kemungkinan klasifikasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=18" title="Sunting kode sumber bagian: Konsep grup Lie, dan kemungkinan klasifikasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grup Lie dianggap sebagai grup kesimetrian yang bervariasi dengan polos. Contoh kesimetrian termasuk rotasi di sekitar sumbu. Yang harus dipahami adalah sifat transformasi 'kecil', misalnya, rotasi melalui sudut-sudut kecil, yang menghubungkan transformasi di dekatnya. Objek matematika yang menangkap struktur ini disebut aljabar Lie (<a href="/wiki/Sophus_Lie" title="Sophus Lie">Lie</a> sendiri menyebutnya "grup infinitesimal"). Dapat didefinisikan karena grup Lie adalah lipatan polos, sehingga memiliki <a href="/w/index.php?title=Ruang_tangen&action=edit&redlink=1" class="new" title="Ruang tangen (halaman belum tersedia)">ruang tangen</a> pada setiap titik. </p><p>Aljabar Lie dari setiap grup Lie kompak (kira-kira: salah satu yang kesimetriannya membentuk himpunan hingga) dapat didekomposisi sebagai <a href="/w/index.php?title=Jumlah_langsung_modul&action=edit&redlink=1" class="new" title="Jumlah langsung modul (halaman belum tersedia)">jumlah langsung</a> dari <a href="/w/index.php?title=Aljabar_Lie_Abelian&action=edit&redlink=1" class="new" title="Aljabar Lie Abelian (halaman belum tersedia)">aljabar Lie Abelian</a> dan sejumlah <a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">sederhana</a>. Struktur aljabar Lie abelian secara matematis tidak menarik, karena tanda kurung Lie identik dengan nol, minatnya terdapat pada ringkasan sederhana. Karenanya muncul pertanyaan, sebagai berikut: Apa <a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">aljabar Lie sederhana</a> dari grup kompak? Ternyata mereka kebanyakan ke dalam empat keluarga tak hingga, "aljabar Lie klasik" A<sub><i>n</i></sub>, B<sub><i>n</i></sub>, C<sub><i>n</i></sub> dan D<sub><i>n</i></sub>, yang dimiliki deskripsi sederhana dalam hal kesimetrian ruang Euklides. Tetapi hanya ada lima "aljabar Lie eksepsional" yang tidak termasuk dalam salah satu keluarga ini. E<sub>8</sub> adalah yang terbesar. </p><p>Grup Lie diklasifikasikan menurut sifat aljabar, yaitu <a href="/wiki/Grup_sederhana" title="Grup sederhana">sederhana</a>, <a href="/w/index.php?title=Grup_semisederhana&action=edit&redlink=1" class="new" title="Grup semisederhana (halaman belum tersedia)">semi-sederhana</a>, <a href="/wiki/Grup_berpenyelesaian" title="Grup berpenyelesaian">berpenyelesaian</a>, <a href="/wiki/Grup_nilpoten" title="Grup nilpoten">nilpoten</a>, <a href="/wiki/Grup_abelian" class="mw-redirect" title="Grup abelian">abelian</a>, <a href="/wiki/Keterhubungan" title="Keterhubungan">keterhubungan</a>, yaitu <a href="/w/index.php?title=Ruang_terkoneksi&action=edit&redlink=1" class="new" title="Ruang terkoneksi (halaman belum tersedia)">terkoneksi</a> atau <a href="/w/index.php?title=Ruang_terkoneksi_sederhana&action=edit&redlink=1" class="new" title="Ruang terkoneksi sederhana (halaman belum tersedia)">terhubung sederhana</a>, dan <a href="/wiki/Ruang_kompak" title="Ruang kompak">kekompakan</a>. </p><p>Hasil utama pertama adalah <a href="/w/index.php?title=Dekomposisi_Levi&action=edit&redlink=1" class="new" title="Dekomposisi Levi (halaman belum tersedia)">dekomposisi Levi</a> yang mengatakan bahwa setiap grup Lie yang terhubung sederhana adalah produk semilangsung dari subgrup normal yang dapat dipecahkan dan subgrup semisederhana. </p> <ul><li><a href="/w/index.php?title=Grup_Lie_kompak&action=edit&redlink=1" class="new" title="Grup Lie kompak (halaman belum tersedia)">Grup Lie kompak</a> yang terhubung yang diketahui: pusat hasil bagi hingga dari produk salinan grup lingkaran <b>S</b><sup>1</sup> dan grup Lie kompak sederhana, yang sesuai dengan <a href="/w/index.php?title=Diagram_Dynkin&action=edit&redlink=1" class="new" title="Diagram Dynkin (halaman belum tersedia)">diagram Dynkin</a> yang terhubung.</li> <li>Setiap gugus Lie berpenyelesaian secara sederhana adalah isomorfik ke subgrup tertutup dari grup matriks segitiga atas invers dari beberapa peringkat, dan wakilan tak tersederhanakan berdimensi-hingga dari grup seperti itu adalah 1-dimensi. Grup berpenyelesaian terlalu berantakan untuk diklasifikasikan kecuali dalam beberapa dimensi kecil.</li> <li>Setiap grup Lie nilpoten yang terhubung sederhana adalah isomorfik ke sungrup tertutup dari grup matriks segitiga atas yang dapat dibalik dengan 1 dalam diagonal dari beberapa peringkat, dan wakilan tak tersederhanakan berdimensi-hingga dari grup adalah 1-dimensi. Seperti grup berpenyelesaian, grup nilpoten untuk diklasifikasikan kecuali dalam beberapa dimensi kecil.</li> <li><a href="/w/index.php?title=Grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Grup Lie sederhana (halaman belum tersedia)">Grup Lie sederhana</a> terkadang didefinisikan sebagai grup yang sederhana sebagai grup abstrak, dan terkadang didefinisikan sebagai grup Lie yang terhubung dengan aljabar Lie sederhana. Misalnya, <a href="/w/index.php?title=SL2(R)&action=edit&redlink=1" class="new" title="SL2(R) (halaman belum tersedia)">SL(2, <b>R</b>)</a> sederhana menurut definisi kedua tetapi tidak menurut definisi pertama. Seluruhnya telah <a href="/w/index.php?title=Daftar_grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Daftar grup Lie sederhana (halaman belum tersedia)">diklasifikasikan</a> (untuk kedua definisi).</li> <li>Grup Lie <a href="/w/index.php?title=Grup_semisederhana&action=edit&redlink=1" class="new" title="Grup semisederhana (halaman belum tersedia)">Semisederhana</a> adalah grup Lie yang aljabar Lie merupakan produk dari aljabar Lie sederhana.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Seluruhnya adalah perluasan utama dari produk grup Lie sederhana.</li></ul> <p><a href="/w/index.php?title=Komponen_identitas&action=edit&redlink=1" class="new" title="Komponen identitas (halaman belum tersedia)">Komponen identitas</a> dari setiap grup Lie adalah <a href="/wiki/Subgrup_normal" title="Subgrup normal">subgrup normal</a> terbuka, dan <a href="/wiki/Grup_hasil_bagi" title="Grup hasil bagi">grup hasil bagi</a> adalah <a href="/w/index.php?title=Grup_diskrit&action=edit&redlink=1" class="new" title="Grup diskrit (halaman belum tersedia)">grup diskrit</a>. Sampul universal dari setiap grup Lie yang terhubung adalah grup Lie yang terhubung secara sederhana, dan sebaliknya setiap grup Lie yang terhubung adalah hasil bagi dari grup Lie yang terhubung secara sederhana oleh subgrup normal diskrit dari pusat. Setiap grup Lie <i>G</i> diuraikan menjadi grup diskrit sederhana, dan abelian dengan cara kanonik sebagai berikut. Ditulis sebagai: </p> <dl><dd><i>G</i><sub>con</sub> untuk komponen identitas yang terhubung</dd> <dd><i>G</i><sub>sol</sub> untuk subgrup berpenyelesaian normal terbesar yang terhubung</dd> <dd><i>G</i><sub>nil</sub> untuk subgrup nilpoten normal terbesar yang terhubung</dd></dl> <p>maka, memiliki urutan subgrup normal </p> <dl><dd>1 ⊆ <i>G</i><sub>nil</sub> ⊆ <i>G</i><sub>sol</sub> ⊆ <i>G</i><sub>con</sub> ⊆ <i>G</i>.</dd></dl> <p>Kemudian </p> <dl><dd><i>G</i>/<i>G</i><sub>con</sub> yang bersifat diskrit</dd> <dd><i>G</i><sub>con</sub>/<i>G</i><sub>sol</sub> adalah <a href="/wiki/Ekstensi_grup" title="Ekstensi grup">ekstensi pusat</a> dari produk <a href="/w/index.php?title=Daftar_grup_Lie_sederhana&action=edit&redlink=1" class="new" title="Daftar grup Lie sederhana (halaman belum tersedia)">grup Lie terhubung sederhana</a>.</dd> <dd><i>G</i><sub>sol</sub>/<i>G</i><sub>nil</sub> yang bersifat abelian. <a href="/w/index.php?title=Grup_Lie_Abelian&action=edit&redlink=1" class="new" title="Grup Lie Abelian (halaman belum tersedia)">Grup Lie Abelian</a> yang terhubung bersifat isomorfik ke produk salinan <b>R</b> dan <a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">grup lingkaran</a> <i>S</i><sup>1</sup>.</dd> <dd><i>G</i><sub>nil</sub>/1 adalah nilpoten, dan oleh karena itu deret pusat menaiknya memiliki semua hasil bagi abelian.</dd></dl> <p>Ini digunakan untuk mengurangi beberapa masalah tentang grup Lie (seperti menemukan wakilan uniter) untuk masalah yang sama untuk grup sederhana yang terhubung dan sungrup nilpoten dan dipecahkan dengan dimensi yang lebih kecil. </p> <ul><li><a href="/w/index.php?title=Difeomorfisme&action=edit&redlink=1" class="new" title="Difeomorfisme (halaman belum tersedia)">Grup difeomorfisme</a> dari grup Lie bertindak secara transitif pada grup Lie</li> <li>Setiap grup Lie adalah <a href="/w/index.php?title=Parallelizabel&action=edit&redlink=1" class="new" title="Parallelizabel (halaman belum tersedia)">parallelizabel</a>, dan karenanya <a href="/w/index.php?title=Lipatan_berorientasi&action=edit&redlink=1" class="new" title="Lipatan berorientasi (halaman belum tersedia)">lipatan berorientasi</a> (terdapat <a href="/w/index.php?title=Berkas_serat&action=edit&redlink=1" class="new" title="Berkas serat (halaman belum tersedia)">isomorfisma berkas</a> antara <a href="/w/index.php?title=Berkas_tangen&action=edit&redlink=1" class="new" title="Berkas tangen (halaman belum tersedia)">berkas tangen</a> dan produk dengan <a href="/w/index.php?title=Ruang_tangen&action=edit&redlink=1" class="new" title="Ruang tangen (halaman belum tersedia)">ruang tangen</a> pada identitasnya)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=19" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=19" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18261910">.mw-parser-output .div-col{margin-top:.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/w/index.php?title=Wakilan_adjoin_dari_grup_Lie&action=edit&redlink=1" class="new" title="Wakilan adjoin dari grup Lie (halaman belum tersedia)">Wakilan adjoin dari grup Lie</a></li> <li><a href="/w/index.php?title=Grup_kompak&action=edit&redlink=1" class="new" title="Grup kompak (halaman belum tersedia)">Grup kompak</a></li> <li><a href="/w/index.php?title=Ukuran_Haar&action=edit&redlink=1" class="new" title="Ukuran Haar (halaman belum tersedia)">Ukuran Haar</a></li> <li><a href="/w/index.php?title=Ruang_homogen&action=edit&redlink=1" class="new" title="Ruang homogen (halaman belum tersedia)">Ruang homogen</a></li> <li><a href="/wiki/Aljabar_Lie" title="Aljabar Lie">Aljabar Lie</a></li> <li><a href="/w/index.php?title=Daftar_topik_grup_Lie&action=edit&redlink=1" class="new" title="Daftar topik grup Lie (halaman belum tersedia)">Daftar topik grup Lie</a></li> <li><a href="/w/index.php?title=Wakilan_grup_Lie&action=edit&redlink=1" class="new" title="Wakilan grup Lie (halaman belum tersedia)">Wakilan grup Lie</a></li> <li><a href="/w/index.php?title=Simetri_dalam_mekanika_kuantum&action=edit&redlink=1" class="new" title="Simetri dalam mekanika kuantum (halaman belum tersedia)">Simetri dalam mekanika kuantum</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Catatan">Catatan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=20" title="Sunting bagian: Catatan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=20" title="Sunting kode sumber bagian: Catatan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Catatan_penjelasan">Catatan penjelasan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=21" title="Sunting bagian: Catatan penjelasan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=21" title="Sunting kode sumber bagian: Catatan penjelasan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18833634">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110928024044/http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf">"Archived copy"</a> <span style="font-size:85%;">(PDF)</span>. Diarsipkan dari <a rel="nofollow" class="external text" href="http://www.math.sunysb.edu/~vkiritch/MAT552/ProblemSet1.pdf">versi asli</a> <span style="font-size:85%;">(PDF)</span> tanggal 2011-09-28<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2014-10-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Archived+copy&rft_id=http%3A%2F%2Fwww.math.sunysb.edu%2F~vkiritch%2FMAT552%2FProblemSet1.pdf&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span> <span style="display:none;font-size:100%" class="error citation-comment">Parameter <code style="color:inherit; border:inherit; padding:inherit;">|url-status=</code> yang tidak diketahui akan diabaikan (<a href="/wiki/Bantuan:Galat_CS1#parameter_ignored" title="Bantuan:Galat CS1">bantuan</a>)</span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Kutipan">Kutipan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=22" title="Sunting bagian: Kutipan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=22" title="Sunting kode sumber bagian: Kutipan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18833634"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Corollary 3.45</span> </li> <li id="cite_note-Hall-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hall_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFRossmann2001">Rossmann 2001</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFT._Kobayashi–T._Oshima">T. Kobayashi–T. Oshima</a>, Definition 5.3.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Ini adalah pernyataan bahwa grup Lie adalah <a href="/w/index.php?title=Grup_Lie_formal&action=edit&redlink=1" class="new" title="Grup Lie formal (halaman belum tersedia)">grup Lie formal</a>. Untuk konsep terakhir, untuk saat ini, lihat F. Bruhat, <a rel="nofollow" class="external text" href="http://www.math.tifr.res.in/~publ/ln/tifr14.pdf">Ceramah tentang Grup Lie dan Grup Wakilan Lokal</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230809143834/http://www.math.tifr.res.in/~publ/ln/tifr14.pdf">Diarsipkan</a> 2023-08-09 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>..</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFRossmann2001">Rossmann 2001</a>, Chapter 2.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Theorem 3.42</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Teorema 5.20</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a> Part III</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation journal">Arthur Tresse (1893). <a rel="nofollow" class="external text" href="https://zenodo.org/record/2273334">"Sur les invariants différentiels des groupes continus de transformations"</a>. <i>Acta Mathematica</i>. <b>18</b>: 1–88. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="plainlinks"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02418270">10.1007/bf02418270</a> <span typeof="mw:File"><span title="Dapat diakses gratis"><img alt="alt=Dapat diakses gratis" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230417161043/https://zenodo.org/record/2273334">Diarsipkan</a> dari versi asli tanggal 2023-04-17<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2021-01-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=Sur+les+invariants+diff%C3%A9rentiels+des+groupes+continus+de+transformations&rft.volume=18&rft.pages=1-88&rft.date=1893&rft_id=info%3Adoi%2F10.1007%2Fbf02418270&rft.au=Arthur+Tresse&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2273334&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-FOOTNOTEBorel2001-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBorel2001_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBorel2001">Borel (2001)</a>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><cite class="citation book">Helgason, Sigurdur (1978). <a rel="nofollow" class="external text" href="https://archive.org/details/differentialgeom00helg_172"><i>Differential Geometry, Lie Groups, and Symmetric Spaces</i></a>. New York: Academic Press. hlm. <a rel="nofollow" class="external text" href="https://archive.org/details/differentialgeom00helg_172/page/n145">131</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-12-338460-7" title="Istimewa:Sumber buku/978-0-12-338460-7">978-0-12-338460-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+Geometry%2C+Lie+Groups%2C+and+Symmetric+Spaces&rft.place=New+York&rft.pages=131&rft.pub=Academic+Press&rft.date=1978&rft.isbn=978-0-12-338460-7&rft.aulast=Helgason&rft.aufirst=Sigurdur&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdifferentialgeom00helg_172&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_Lie&veaction=edit&section=23" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_Lie&action=edit&section=23" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite id="CITEREFAdams1969" class="citation"><a href="/w/index.php?title=John_Frank_Adams&action=edit&redlink=1" class="new" title="John Frank Adams (halaman belum tersedia)">Adams, John Frank</a> (1969), <i>Lectures on Lie Groups</i>, Chicago Lectures in Mathematics, Chicago: Univ. of Chicago Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-226-00527-0" title="Istimewa:Sumber buku/978-0-226-00527-0">978-0-226-00527-0</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0252560">0252560</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Lie+Groups&rft.place=Chicago&rft.series=Chicago+Lectures+in+Mathematics&rft.pub=Univ.+of+Chicago+Press&rft.date=1969&rft.isbn=978-0-226-00527-0&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0252560&rft.aulast=Adams&rft.aufirst=John+Frank&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite class="citation book">Bäuerle, G.G.A; de Kerf, E.A.; ten Kroode, A. P. E. (1997). A. van Groesen; E.M. de Jager, ed. <span class="plainlinks"><a rel="nofollow" class="external text" href="http://www.sciencedirect.com/science/bookseries/09258582"><i>Finite and infinite dimensional Lie algebras and their application in physics</i><span style="padding-left:0.15em"><span typeof="mw:File"><span title="Perlu langganan berbayar"><img alt="Perlu langganan berbayar" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Lock-red-L.svg/9px-Lock-red-L.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Lock-red-L.svg/14px-Lock-red-L.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Lock-red-L.svg/18px-Lock-red-L.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span></a></span>. Studies in mathematical physics. <b>7</b>. North-Holland. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-444-82836-1" title="Istimewa:Sumber buku/978-0-444-82836-1">978-0-444-82836-1</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170704154138/http://www.sciencedirect.com/science/bookseries/09258582">Diarsipkan</a> dari versi asli tanggal 2017-07-04<span class="reference-accessdate">. Diakses tanggal <span class="nowrap">2021-01-01</span></span> – via <a href="/wiki/ScienceDirect" title="ScienceDirect">ScienceDirect</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+and+infinite+dimensional+Lie+algebras+and+their+application+in+physics&rft.series=Studies+in+mathematical+physics&rft.pub=North-Holland&rft.date=1997&rft.isbn=978-0-444-82836-1&rft.aulast=B%C3%A4uerle&rft.aufirst=G.G.A&rft.au=de+Kerf%2C+E.A.&rft.au=ten+Kroode%2C+A.+P.+E.&rft_id=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%2Fbookseries%2F09258582&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFBorel2001" class="citation"><a href="/w/index.php?title=Armand_Borel&action=edit&redlink=1" class="new" title="Armand Borel (halaman belum tersedia)">Borel, Armand</a> (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?isbn=0821802887"><i>Essays in the history of Lie groups and algebraic groups</i></a>, History of Mathematics, <b>21</b>, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-8218-0288-5" title="Istimewa:Sumber buku/978-0-8218-0288-5">978-0-8218-0288-5</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1847105">1847105</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+in+the+history+of+Lie+groups+and+algebraic+groups&rft.place=Providence%2C+R.I.&rft.series=History+of+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0-8218-0288-5&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1847105&rft.aulast=Borel&rft.aufirst=Armand&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fisbn%3D0821802887&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFBourbaki" class="citation"><a href="/w/index.php?title=Nicolas_Bourbaki&action=edit&redlink=1" class="new" title="Nicolas Bourbaki (halaman belum tersedia)">Bourbaki, Nicolas</a>, <i>Elements of mathematics: Lie groups and Lie algebras</i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+mathematics%3A+Lie+groups+and+Lie+algebras&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>. Chapters 1–3 <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/3-540-64242-0" title="Istimewa:Sumber buku/3-540-64242-0">3-540-64242-0</a>, Chapters 4–6 <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/3-540-42650-7" title="Istimewa:Sumber buku/3-540-42650-7">3-540-42650-7</a>, Chapters 7–9 <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/3-540-43405-4" title="Istimewa:Sumber buku/3-540-43405-4">3-540-43405-4</a></li> <li><cite id="CITEREFChevalley1946" class="citation">Chevalley, Claude (1946), <i>Theory of Lie groups</i>, Princeton: Princeton University Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-691-04990-8" title="Istimewa:Sumber buku/978-0-691-04990-8">978-0-691-04990-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Lie+groups&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=1946&rft.isbn=978-0-691-04990-8&rft.aulast=Chevalley&rft.aufirst=Claude&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><a href="/w/index.php?title=P._M._Cohn&action=edit&redlink=1" class="new" title="P. M. Cohn (halaman belum tersedia)">P. M. Cohn</a> (1957) <i>Lie Groups</i>, Cambridge Tracts in Mathematical Physics.</li> <li><a href="/w/index.php?title=J._L._Coolidge&action=edit&redlink=1" class="new" title="J. L. Coolidge (halaman belum tersedia)">J. L. Coolidge</a> (1940) <i>A History of Geometrical Methods</i>, pp 304–17, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a> (<a href="/w/index.php?title=Dover_Publications&action=edit&redlink=1" class="new" title="Dover Publications (halaman belum tersedia)">Dover Publications</a> 2003).</li> <li><a href="/w/index.php?title=Templat:Fulton-Harris&action=edit&redlink=1" class="new" title="Templat:Fulton-Harris (halaman belum tersedia)">Templat:Fulton-Harris</a></li> <li>Robert Gilmore (2008) <i>Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/9780521884006" title="Istimewa:Sumber buku/9780521884006">9780521884006</a> <a href="/wiki/Pengenal_objek_digital" title="Pengenal objek digital">DOI</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/CBO9780511791390">10.1017/CBO9780511791390</a>.</li> <li><cite id="CITEREFHall2015" class="citation">Hall, Brian C. (2015), <i>Lie Groups, Lie Algebras, and Representations: An Elementary Introduction</i>, Graduate Texts in Mathematics, <b>222</b> (edisi ke-2nd), Springer, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-319-13467-3">10.1007/978-3-319-13467-3</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-3319134666" title="Istimewa:Sumber buku/978-3319134666">978-3319134666</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%2C+Lie+Algebras%2C+and+Representations%3A+An+Elementary+Introduction&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2015&rft_id=info%3Adoi%2F10.1007%2F978-3-319-13467-3&rft.isbn=978-3319134666&rft.aulast=Hall&rft.aufirst=Brian+C.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li>F. Reese Harvey (1990) <i>Spinors and calibrations</i>, <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/0-12-329650-1" title="Istimewa:Sumber buku/0-12-329650-1">0-12-329650-1</a>.</li> <li><cite id="CITEREFHawkins2000" class="citation">Hawkins, Thomas (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?isbn=978-0-387-98963-1"><i>Emergence of the theory of Lie groups</i></a>, Sources and Studies in the History of Mathematics and Physical Sciences, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="plainlinks"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-1202-7">10.1007/978-1-4612-1202-7</a> <span typeof="mw:File"><span title="Dapat diakses gratis"><img alt="alt=Dapat diakses gratis" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-98963-1" title="Istimewa:Sumber buku/978-0-387-98963-1">978-0-387-98963-1</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1771134">1771134</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Emergence+of+the+theory+of+Lie+groups&rft.place=Berlin%2C+New+York&rft.series=Sources+and+Studies+in+the+History+of+Mathematics+and+Physical+Sciences&rft.pub=Springer-Verlag&rft.date=2000&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1771134&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-1202-7&rft.isbn=978-0-387-98963-1&rft.aulast=Hawkins&rft.aufirst=Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fisbn%3D978-0-387-98963-1&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2695575">Borel's review</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190427090037/https://www.jstor.org/stable/2695575">Diarsipkan</a> 2019-04-27 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><cite id="CITEREFHelgason2001" class="citation"><a href="/w/index.php?title=Sigur%C3%B0ur_Helgason_(mathematician)&action=edit&redlink=1" class="new" title="Sigurður Helgason (mathematician) (halaman belum tersedia)">Helgason, Sigurdur</a> (2001), <i>Differential geometry, Lie groups, and symmetric spaces</i>, Graduate Studies in Mathematics, <b>34</b>, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fgsm%2F034">10.1090/gsm/034</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-8218-2848-9" title="Istimewa:Sumber buku/978-0-8218-2848-9">978-0-8218-2848-9</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1834454">1834454</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differential+geometry%2C+Lie+groups%2C+and+symmetric+spaces&rft.place=Providence%2C+R.I.&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2001&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1834454&rft_id=info%3Adoi%2F10.1090%2Fgsm%2F034&rft.isbn=978-0-8218-2848-9&rft.aulast=Helgason&rft.aufirst=Sigurdur&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFKnapp2002" class="citation"><a href="/w/index.php?title=Anthony_Knapp&action=edit&redlink=1" class="new" title="Anthony Knapp (halaman belum tersedia)">Knapp, Anthony W.</a> (2002), <i>Lie Groups Beyond an Introduction</i>, Progress in Mathematics, <b>140</b> (edisi ke-2nd), Boston: Birkhäuser, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-8176-4259-4" title="Istimewa:Sumber buku/978-0-8176-4259-4">978-0-8176-4259-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups+Beyond+an+Introduction&rft.place=Boston&rft.series=Progress+in+Mathematics&rft.edition=2nd&rft.pub=Birkh%C3%A4user&rft.date=2002&rft.isbn=978-0-8176-4259-4&rft.aulast=Knapp&rft.aufirst=Anthony+W.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li>T. Kobayashi and T. Oshima, Lie groups and Lie algebras I, Iwanami, 1999 (in Japanese)</li> <li><cite class="citation journal"><a href="/w/index.php?title=Albert_Nijenhuis&action=edit&redlink=1" class="new" title="Albert Nijenhuis (halaman belum tersedia)">Nijenhuis, Albert</a> (1959). "Review: <i>Lie groups</i>, by P. M. Cohn". <i><a href="/w/index.php?title=Bulletin_of_the_American_Mathematical_Society&action=edit&redlink=1" class="new" title="Bulletin of the American Mathematical Society (halaman belum tersedia)">Bulletin of the American Mathematical Society</a></i>. <b>65</b> (6): 338–341. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<span class="plainlinks"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1959-10358-x">10.1090/s0002-9904-1959-10358-x</a> <span typeof="mw:File"><span title="Dapat diakses gratis"><img alt="alt=Dapat diakses gratis" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Review%3A+Lie+groups%2C+by+P.+M.+Cohn&rft.volume=65&rft.issue=6&rft.pages=338-341&rft.date=1959&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1959-10358-x&rft.au=Nijenhuis%2C+Albert&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFRossmann2001" class="citation">Rossmann, Wulf (2001), <i>Lie Groups: An Introduction Through Linear Groups</i>, Oxford Graduate Texts in Mathematics, Oxford University Press, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-19-859683-7" title="Istimewa:Sumber buku/978-0-19-859683-7">978-0-19-859683-7</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%3A+An+Introduction+Through+Linear+Groups&rft.series=Oxford+Graduate+Texts+in+Mathematics&rft.pub=Oxford+University+Press&rft.date=2001&rft.isbn=978-0-19-859683-7&rft.aulast=Rossmann&rft.aufirst=Wulf&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>. The 2003 reprint corrects several typographical mistakes.</li> <li><cite class="citation book">Sattinger, David H.; Weaver, O. L. (1986). <i>Lie groups and algebras with applications to physics, geometry, and mechanics</i>. Springer-Verlag. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-1910-9">10.1007/978-1-4757-1910-9</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-3-540-96240-3" title="Istimewa:Sumber buku/978-3-540-96240-3">978-3-540-96240-3</a>. <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0835009">0835009</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+groups+and+algebras+with+applications+to+physics%2C+geometry%2C+and+mechanics&rft.pub=Springer-Verlag&rft.date=1986&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0835009&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-1910-9&rft.isbn=978-3-540-96240-3&rft.aulast=Sattinger&rft.aufirst=David+H.&rft.au=Weaver%2C+O.+L.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFSerre1965" class="citation"><a href="/w/index.php?title=J.-P._Serre&action=edit&redlink=1" class="new" title="J.-P. Serre (halaman belum tersedia)">Serre, Jean-Pierre</a> (1965), <i>Lie Algebras and Lie Groups: 1964 Lectures given at Harvard University</i>, Lecture notes in mathematics, <b>1500</b>, Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-3-540-55008-2" title="Istimewa:Sumber buku/978-3-540-55008-2">978-3-540-55008-2</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Algebras+and+Lie+Groups%3A+1964+Lectures+given+at+Harvard+University&rft.series=Lecture+notes+in+mathematics&rft.pub=Springer&rft.date=1965&rft.isbn=978-3-540-55008-2&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite class="citation book"><a href="/w/index.php?title=John_Stillwell&action=edit&redlink=1" class="new" title="John Stillwell (halaman belum tersedia)">Stillwell, John</a> (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/naivelietheory0000stil"><i>Naive Lie Theory</i></a>. Undergraduate Texts in Mathematics. Springer. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-78214-0">10.1007/978-0-387-78214-0</a>. <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0387782140" title="Istimewa:Sumber buku/978-0387782140">978-0387782140</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Lie+Theory&rft.series=Undergraduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-0-387-78214-0&rft.isbn=978-0387782140&rft.aulast=Stillwell&rft.aufirst=John&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnaivelietheory0000stil&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li>Heldermann Verlag <a rel="nofollow" class="external text" href="http://www.heldermann.de/JLT/jltcover.htm">Journal of Lie Theory</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230608104226/https://www.heldermann.de/JLT/jltcover.htm">Diarsipkan</a> 2023-06-08 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</li> <li><cite id="CITEREFWarner1983" class="citation">Warner, Frank W. (1983), <i>Foundations of differentiable manifolds and Lie groups</i>, Graduate Texts in Mathematics, <b>94</b>, New York Berlin Heidelberg: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-1799-0">10.1007/978-1-4757-1799-0</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-90894-6" title="Istimewa:Sumber buku/978-0-387-90894-6">978-0-387-90894-6</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0722297">0722297</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+differentiable+manifolds+and+Lie+groups&rft.place=New+York+Berlin+Heidelberg&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1983&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0722297&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-1799-0&rft.isbn=978-0-387-90894-6&rft.aulast=Warner&rft.aufirst=Frank+W.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFSteeb2007" class="citation">Steeb, Willi-Hans (2007), <i>Continuous Symmetries, Lie algebras, Differential Equations and Computer Algebra: second edition</i>, World Scientific Publishing, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F6515">10.1142/6515</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-981-270-809-0" title="Istimewa:Sumber buku/978-981-270-809-0">978-981-270-809-0</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=2382250">2382250</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Continuous+Symmetries%2C+Lie+algebras%2C+Differential+Equations+and+Computer+Algebra%3A+second+edition&rft.pub=World+Scientific+Publishing&rft.date=2007&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D2382250&rft_id=info%3Adoi%2F10.1142%2F6515&rft.isbn=978-981-270-809-0&rft.aulast=Steeb&rft.aufirst=Willi-Hans&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+Lie" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><a rel="nofollow" class="external text" href="http://www.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf">Lie Groups. Representation Theory and Symmetric Spaces</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20230416100215/https://www2.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf">Diarsipkan</a> 2023-04-16 di <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Wolfgang Ziller, Vorlesung 2010</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><style data-mw-deduplicate="TemplateStyles:r25847331">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output 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