Unipotent — Wikipédia

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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Définition_avec_la_théorie_des_représentations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Définition avec la théorie des représentations</span> </div> </a> <ul id="toc-Définition_avec_la_théorie_des_représentations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exemples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemples</span> </div> </a> <button aria-controls="toc-Exemples-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>Afficher / masquer la sous-section Exemples</span> </button> <ul id="toc-Exemples-sublist" class="vector-toc-list"> <li id="toc-Un" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Un"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>U<sub><i>n</i></sub></span> </div> </a> <ul id="toc-Un-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-(Ga)n" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#(Ga)n"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>(G<sub>a</sub>)<sup><i>n</i></sup></span> </div> </a> <ul id="toc-(Ga)n-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noyau_du_Frobenius" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noyau_du_Frobenius"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Noyau du Frobenius</span> </div> </a> <ul id="toc-Noyau_du_Frobenius-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Classification_des_groupes_unipotents_en_caractéristique_0" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_des_groupes_unipotents_en_caractéristique_0"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Classification des groupes unipotents en caractéristique 0</span> </div> </a> <button aria-controls="toc-Classification_des_groupes_unipotents_en_caractéristique_0-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>Afficher / masquer la sous-section Classification des groupes unipotents en caractéristique 0</span> </button> <ul id="toc-Classification_des_groupes_unipotents_en_caractéristique_0-sublist" class="vector-toc-list"> <li id="toc-Remarques" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Remarques"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Remarques</span> </div> </a> <ul id="toc-Remarques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Radical_unipotent" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Radical_unipotent"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Radical unipotent</span> </div> </a> <ul id="toc-Radical_unipotent-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Décomposition_des_groupes_algébriques" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Décomposition_des_groupes_algébriques"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Décomposition des groupes algébriques</span> </div> </a> <button aria-controls="toc-Décomposition_des_groupes_algébriques-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>Afficher / masquer la sous-section Décomposition des groupes algébriques</span> </button> <ul id="toc-Décomposition_des_groupes_algébriques-sublist" class="vector-toc-list"> <li id="toc-Caractéristique_0" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caractéristique_0"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Caractéristique 0</span> </div> </a> <ul id="toc-Caractéristique_0-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Caractéristique_p" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Caractéristique_p"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Caractéristique <i>p</i></span> </div> </a> <ul id="toc-Caractéristique_p-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Décomposition_de_Jordan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Décomposition_de_Jordan"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Décomposition de Jordan</span> </div> </a> <ul id="toc-Décomposition_de_Jordan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Voir aussi</span> </div> </a> <button aria-controls="toc-Voir_aussi-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>Afficher / masquer la sous-section Voir aussi</span> </button> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> <li id="toc-Bibliographie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliographie"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Bibliographie</span> </div> </a> <ul id="toc-Bibliographie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Articles_connexes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Articles_connexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Articles connexes</span> </div> </a> <ul id="toc-Articles_connexes-sublist" class="vector-toc-list"> </ul> </li> </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 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Unipotent</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="Aller à un article dans une autre langue. Disponible en 6 langues." > <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-6" 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">6 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Unipotentes_Element" title="Unipotentes Element – allemand" lang="de" hreflang="de" data-title="Unipotentes Element" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%BF%CE%BD%CE%BF%CE%B4%CF%8D%CE%BD%CE%B1%CE%BC%CE%BF_%CF%83%CF%84%CE%BF%CE%B9%CF%87%CE%B5%CE%AF%CE%BF" title="Μονοδύναμο στοιχείο – grec" lang="el" hreflang="el" data-title="Μονοδύναμο στοιχείο" data-language-autonym="Ελληνικά" data-language-local-name="grec" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Unipotent" title="Unipotent – anglais" lang="en" hreflang="en" data-title="Unipotent" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A9%B1%EC%9D%BC%EC%9B%90" title="멱일원 – coréen" lang="ko" hreflang="ko" data-title="멱일원" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Unipotentnost" title="Unipotentnost – slovène" lang="sl" hreflang="sl" data-title="Unipotentnost" data-language-autonym="Slovenščina" data-language-local-name="slovène" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A3%D0%BD%D1%96%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%82%D0%BD%D0%B8%D0%B9_%D0%B5%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82" title="Уніпотентний елемент – ukrainien" lang="uk" hreflang="uk" data-title="Уніпотентний елемент" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</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/Q2494915#sitelinks-wikipedia" title="Modifier les liens 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vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Unipotent&action=edit" title="Modifier le wikicode de cette page [e]" accesskey="e"><span>Modifier le code</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Unipotent&action=history"><span>Voir l’historique</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Général </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_li%C3%A9es/Unipotent" title="Liste des pages liées qui pointent sur celle-ci [j]" accesskey="j"><span>Pages liées</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Suivi_des_liens/Unipotent" rel="nofollow" title="Liste des modifications récentes des pages appelées par celle-ci [k]" accesskey="k"><span>Suivi des pages liées</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Aide:Importer_un_fichier" title="Téléverser des fichiers [u]" accesskey="u"><span>Téléverser un fichier</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Sp%C3%A9cial:Pages_sp%C3%A9ciales" title="Liste de toutes les pages spéciales [q]" accesskey="q"><span>Pages spéciales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Unipotent&oldid=207738385" title="Adresse permanente de cette version de cette page"><span>Lien permanent</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Unipotent&action=info" title="Davantage d’informations sur cette page"><span>Informations sur la page</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:Citer&page=Unipotent&id=207738385&wpFormIdentifier=titleform" title="Informations sur la manière de citer cette page"><span>Citer cette page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:UrlShortener&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FUnipotent"><span>Obtenir l'URL raccourcie</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:QrCode&url=https%3A%2F%2Ffr.wikipedia.org%2Fwiki%2FUnipotent"><span>Télécharger le code 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"> Imprimer / exporter </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=Sp%C3%A9cial:Livre&bookcmd=book_creator&referer=Unipotent"><span>Créer un livre</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Sp%C3%A9cial:DownloadAsPdf&page=Unipotent&action=show-download-screen"><span>Télécharger comme PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Unipotent&printable=yes" title="Version imprimable de cette page [p]" accesskey="p"><span>Version imprimable</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dans d’autres projets </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2494915" title="Lien vers l’élément dans le dépôt de données connecté [g]" accesskey="g"><span>Élément 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="Outils de la page"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apparence"> <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">Apparence</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</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">Un article de Wikipédia, l'encyclopédie libre.</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="fr" dir="ltr"><p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, un <b>élément unipotent</b> <i>r</i> d'un <a href="/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques)">anneau</a> unitaire <i>R</i> est un tel que <i>r </i>− 1 est un <a href="/wiki/Nilpotent" title="Nilpotent">élément nilpotent</a> ; en d'autres termes, (<i>r</i> − 1)<sup><i>n</i></sup> vaut zéro pour <i>n</i> assez grand. </p><p>En particulier, une matrice carrée <i>M</i> est une <b>matrice unipotente</b> si et seulement si son <a href="/wiki/Polyn%C3%B4me_caract%C3%A9ristique" title="Polynôme caractéristique">polynôme caractéristique</a> <i>P</i>(<i>t</i>) est une puissance de <i>t</i> − 1. Ainsi, toutes les <a href="/wiki/Valeur_propre,_vecteur_propre_et_espace_propre" title="Valeur propre, vecteur propre et espace propre">valeurs propres</a> d'une matrice unipotente valent 1. </p><p>Le terme <b>quasi-unipotent</b> signifie qu'une certaine puissance de l'élément est unipotente. Par exemple, une <a href="/wiki/Matrice_diagonalisable" title="Matrice diagonalisable">matrice diagonalisable</a> dont toutes les valeurs propres sont des <a href="/wiki/Racine_de_l%27unit%C3%A9" title="Racine de l'unité">racines de l'unité</a> est quasi-unipotente. </p><p>Dans la théorie des <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">groupes algébriques</a>, un élément d'un groupe est <b>unipotent</b> s'il agit de manière unipotente dans une certaine <a href="/wiki/Repr%C3%A9sentation_de_groupe" title="Représentation de groupe">représentation naturelle du groupe</a>. Un <b>groupe algébrique affine unipotent</b> est alors un groupe dont tous les éléments sont unipotents. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=1" title="Modifier la section : Définition" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=1" title="Modifier le code source de la section : Définition"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Définition_avec_des_matrices"><span id="D.C3.A9finition_avec_des_matrices"></span>Définition avec des matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=2" title="Modifier la section : Définition avec des matrices" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=2" title="Modifier le code source de la section : Définition avec des matrices"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour <i>n</i> <a href="/wiki/Entier_naturel" title="Entier naturel">entier naturel</a>, soit <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 {U} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c30194acc8d4860819d47d0a9e3845449a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}"></span> le <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> des <a href="/wiki/Matrice_triangulaire" title="Matrice triangulaire">matrices triangulaires supérieures</a> avec des <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> sur la diagonale, c'est-à-dire le groupe<sup id="cite_ref-M252_1-0" class="reference"><a href="#cite_note-M252-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> </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 \mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01c89d341a4bb14b0bc4cc4438c42f0818a5192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:31.676ex; height:16.843ex;" alt="{\displaystyle \mathbb {U} _{n}=\left\{{\begin{bmatrix}1&*&\cdots &*&*\\0&1&\cdots &*&*\\\vdots &\vdots &&\vdots &\vdots \\0&0&\cdots &1&*\\0&0&\cdots &0&1\end{bmatrix}}\right\}.}"></span></dd></dl> <p>Alors, un <b>groupe unipotent</b> peut être défini comme étant un groupe isomorphe à un <a href="/wiki/Sous-groupe" title="Sous-groupe">sous-groupe</a> d'un certain <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 {U} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c30194acc8d4860819d47d0a9e3845449a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}"></span>. En utilisant la <a href="/wiki/Sch%C3%A9ma_(g%C3%A9om%C3%A9trie_alg%C3%A9brique)" title="Schéma (géométrie algébrique)">théorie des schémas</a>, le groupe <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 {U} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c30194acc8d4860819d47d0a9e3845449a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}"></span> peut être défini comme le schéma en groupes </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 {\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mtext>det</mtext> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>></mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce88e4038dfda44d7929124d20ee7522355e21df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.125ex; height:10.176ex;" alt="{\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} \!\left[x_{11},x_{12},\ldots ,x_{nn},{\frac {1}{\text{det}}}\right]}{(x_{ii}=1,x_{i>j}=0)}}\right)}"></span></dd></dl> <p>et un schéma en groupes affine est unipotent s'il est isomorphe à un sous-schéma en groupes fermé de ce schéma. </p> <div class="mw-heading mw-heading3"><h3 id="Définition_avec_la_théorie_des_anneaux"><span id="D.C3.A9finition_avec_la_th.C3.A9orie_des_anneaux"></span>Définition avec la théorie des anneaux</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=3" title="Modifier la section : Définition avec la théorie des anneaux" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=3" title="Modifier le code source de la section : Définition avec la théorie des anneaux"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un élément <i>x</i> d'un <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">groupe algébrique</a> affine <i>G</i> est unipotent si l'opérateur de translation à droite associé, <i>r</i><sub><i>x</i></sub>, sur l'<a href="/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique_affine" title="Variété algébrique affine">anneau de coordonnées affines</a> <i>A</i>[<i>G</i>] de <i>G</i> est localement unipotent en tant qu'élément de l'anneau des <a href="/wiki/Application_lin%C3%A9aire" title="Application linéaire">endomorphismes linéaires</a> de <i>A</i>[<i>G</i>]. (Ici, « localement unipotent » signifie que la restriction à tout sous-espace stable de dimension finie de <i>A</i>[<i>G</i>] est unipotente au sens habituel de la théorie des anneaux.) </p><p>Un groupe algébrique affine est dit <b>unipotent</b> si tous ses éléments sont unipotents. Tout groupe algébrique unipotent est <a href="/wiki/Isomorphisme" title="Isomorphisme">isomorphe</a> à un sous-groupe fermé du groupe des matrices triangulaires supérieures dont les coefficients diagonaux valent 1, et inversement tout tel sous-groupe est unipotent. En particulier tout groupe unipotent est un <a href="/wiki/Groupe_nilpotent" title="Groupe nilpotent">groupe nilpotent</a>, bien que l'inverse ne soit pas vrai (contre-exemple : le groupe des <a href="/wiki/Matrice_diagonale" title="Matrice diagonale">matrices diagonales</a> de GL<sub><i>n</i></sub>(<i>k</i>) est nilpotent puisqu'il est abélien mais il n'est pas unipotent). </p><p>Par exemple, la représentation standard de <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 {U} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c30194acc8d4860819d47d0a9e3845449a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}"></span> sur <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 k^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d0ca5fd176db2867ec07a961a31f17bc6fb07e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.43ex; height:2.343ex;" alt="{\displaystyle k^{n}}"></span> avec base standard <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_{i})_{1\leq i\leq n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (e_{i})_{1\leq i\leq n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd948ce791e58bb05ad024fa21b033ee8b858ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.857ex; height:2.843ex;" alt="{\displaystyle (e_{i})_{1\leq i\leq n}}"></span> admet pour vecteur fixe <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_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e81caf3d4bcb929315801cbabc83543829484ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle e_{1}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Définition_avec_la_théorie_des_représentations"><span id="D.C3.A9finition_avec_la_th.C3.A9orie_des_repr.C3.A9sentations"></span>Définition avec la théorie des représentations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=4" title="Modifier la section : Définition avec la théorie des représentations" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=4" title="Modifier le code source de la section : Définition avec la théorie des représentations"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si un groupe unipotent agit sur une <a href="/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique_affine" title="Variété algébrique affine">variété affine</a>, toutes ses orbites sont fermées, et s'il agit linéairement sur un <a href="/wiki/Espace_vectoriel" title="Espace vectoriel">espace vectoriel</a> de dimension finie, alors il admet un vecteur fixe non nul. En fait, cette dernière propriété caractérise les groupes unipotents<sup id="cite_ref-M252_1-1" class="reference"><a href="#cite_note-M252-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. En particulier, cela implique qu'il n'y a pas de représentations semi-simples non triviales. </p> <div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=5" title="Modifier la section : Exemples" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=5" title="Modifier le code source de la section : Exemples"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Un">U<sub><i>n</i></sub></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=6" title="Modifier la section : Un" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=6" title="Modifier le code source de la section : Un"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bien sûr, le groupe de matrices <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 {U} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e86c30194acc8d4860819d47d0a9e3845449a850" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \mathbb {U} _{n}}"></span> est unipotent. En utilisant la série centrale descendante </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 \mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>⊃</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>⊃</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>⊃</mo> <mo>⋯</mo> <mo>⊃</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mi>e</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08b1d4a99066365a9415a4043e3a1dc274ff44cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:42.608ex; height:3.343ex;" alt="{\displaystyle \mathbb {U} _{n}=\mathbb {U} _{n}^{(0)}\supset \mathbb {U} _{n}^{(1)}\supset \mathbb {U} _{n}^{(2)}\supset \cdots \supset \mathbb {U} _{n}^{(m)}=e,}"></span></dd></dl> <p>où </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 \mathbb {U} _{n}^{(1)}=[\mathbb {U} _{n},\mathbb {U} _{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}^{(1)}=[\mathbb {U} _{n},\mathbb {U} _{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f416a069728bba921863fa4f0aae18603cd206" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.231ex; height:3.509ex;" alt="{\displaystyle \mathbb {U} _{n}^{(1)}=[\mathbb {U} _{n},\mathbb {U} _{n}]}"></span> et <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 {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa66ccfb6bdc85c972ec663793bd3752780f4c8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.993ex; height:3.509ex;" alt="{\displaystyle \mathbb {U} _{n}^{(2)}=[\mathbb {U} _{n},\mathbb {U} _{n}^{(1)}],}"></span> etc.,</dd></dl> <p>on voit apparaître des groupes unipotents. Par exemple, pour <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=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d928ec15aeef83aade867992ee473933adb6139d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=4}"></span>, les séries centrales sont les groupes matriciels </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 \mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2645d2ea1b3ee40bb54b27bdd7709d606428a071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:25.432ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}=\left\{{\begin{bmatrix}1&*&*&*\\0&1&*&*\\0&0&1&*\\0&0&0&1\end{bmatrix}}\right\}}"></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 \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>∗</mo> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4deea0c44875459cce0ed9e7500511581b0cb17b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.712ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(1)}=\left\{{\begin{bmatrix}1&0&*&*\\0&1&0&*\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}"></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 \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>∗</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55d073e2d49a30f86097bca803348ad0d3e701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:26.712ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(2)}=\left\{{\begin{bmatrix}1&0&0&*\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\}}"></span>, et <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 {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a62e5c53f681d8cf2a9d683fd43d44caf155d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:27.746ex; height:12.509ex;" alt="{\displaystyle \mathbb {U} _{4}^{(3)}=\left\{{\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}\right\},}"></span></dd></dl> <p>ce qui donne quelques exemples de groupes unipotents. </p> <div class="mw-heading mw-heading3"><h3 id="(Ga)n"><span id=".28Ga.29n"></span>(G<sub>a</sub>)<sup><i>n</i></sup></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=7" title="Modifier la section : (Ga)n" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=7" title="Modifier le code source de la section : (Ga)n"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On voit que le groupe additif <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 {G} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a118277795dec18b43d8e6b839e6af5856180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}"></span> est un groupe unipotent grâce à l'injection </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 a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2926a27d2831959050d3d669f651fe62ef493da8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.412ex; height:6.176ex;" alt="{\displaystyle a\mapsto {\begin{bmatrix}1&a\\0&1\end{bmatrix}}.}"></span></dd></dl> <p>Remarquons que la multiplication matricielle donne </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 {\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11fd0a315c167ad04f6c723fa72c5a1af847ee88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.959ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&b\\0&1\end{bmatrix}}={\begin{bmatrix}1&a+b\\0&1\end{bmatrix}},}"></span></dd></dl> <p>de sorte qu'il s'agit bien d'un <a href="/wiki/Morphisme_de_groupes" title="Morphisme de groupes">morphisme de groupes</a> (injectif). Plus généralement, on obtient un morphisme injectif <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 {G} _{a}^{n}\to \mathbb {U} _{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">U</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea94030e7ca5d536cd159824e7267a06e9579032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.638ex; height:2.843ex;" alt="{\displaystyle \mathbb {G} _{a}^{n}\to \mathbb {U} _{n+1}}"></span> grâce à l'application suivante : </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 (a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&a_{n}\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\\0&0&0&\cdots &0&1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&a_{n}\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\\0&0&0&\cdots &0&1\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96697417c8af6155ca8f9c8294bb26da8e107a66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:47.284ex; height:17.176ex;" alt="{\displaystyle (a_{1},\ldots ,a_{n})\,\mapsto {\begin{bmatrix}1&a_{1}&a_{2}&\cdots &a_{n-1}&a_{n}\\0&1&0&\cdots &0&0\\\vdots &\vdots &\vdots &&\vdots &\vdots \\0&0&0&\cdots &1&0\\0&0&0&\cdots &0&1\end{bmatrix}}.}"></span></dd></dl> <p>En termes de théorie des schémas, <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 {G} _{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {G} _{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a118277795dec18b43d8e6b839e6af5856180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.91ex; height:2.509ex;" alt="{\displaystyle \mathbb {G} _{a}}"></span> est donnée par le <a href="/wiki/Foncteur" title="Foncteur">foncteur</a> </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 {\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Sch</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>p</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Sets</mtext> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d14c9e9d7e606c423ce16dd7c090dab3dffe19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:18.222ex; height:2.343ex;" alt="{\displaystyle {\mathcal {O}}:{\textbf {Sch}}^{op}\to {\textbf {Sets}}}"></span></dd></dl> <p>défini par </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,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/175fc598b912719478c67e3cf85f03f43257bb67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.838ex; height:2.843ex;" alt="{\displaystyle (X,{\mathcal {O}}_{X})\mapsto {\mathcal {O}}_{X}(X).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Noyau_du_Frobenius">Noyau du Frobenius</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=8" title="Modifier la section : Noyau du Frobenius" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=8" title="Modifier le code source de la section : Noyau du Frobenius"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Partant du foncteur <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 {\mathcal {O}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ae2ed4058fb748a183d9ada8aea50a00d0c89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.85ex; height:2.176ex;" alt="{\displaystyle {\mathcal {O}}}"></span> sur la <a href="/wiki/Sous-cat%C3%A9gorie" class="mw-redirect" title="Sous-catégorie">sous-catégorie</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 {\textbf {Sch}}/\mathbb {F} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">Sch</mtext> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textbf {Sch}}/\mathbb {F} _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/799429666a55d6760bdf6956ee92d6097342a4b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.8ex; height:3.009ex;" alt="{\displaystyle {\textbf {Sch}}/\mathbb {F} _{p}}"></span>, on a le sous-foncteur <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 \alpha _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c48aa9000af59f94d3022f58beadb61cea7d8b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.547ex; height:2.343ex;" alt="{\displaystyle \alpha _{p}}"></span> défini par </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 \alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690844dcdeaa6d248fe00977dff17ecfd7672cb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.156ex; height:3.009ex;" alt="{\displaystyle \alpha _{p}(X)=\{x\in {\mathcal {O}}(X):x^{p}=0\}}"></span>.</dd></dl> <p>Il permet de définir un groupe unipotent, le noyau de l'<a href="/wiki/Endomorphisme_de_Frobenius" title="Endomorphisme de Frobenius">endomorphisme de Frobenius</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_des_groupes_unipotents_en_caractéristique_0"><span id="Classification_des_groupes_unipotents_en_caract.C3.A9ristique_0"></span>Classification des groupes unipotents en caractéristique 0</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=9" title="Modifier la section : Classification des groupes unipotents en caractéristique 0" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=9" title="Modifier le code source de la section : Classification des groupes unipotents en caractéristique 0"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En <a href="/wiki/Caract%C3%A9ristique_d%27un_anneau" title="Caractéristique d'un anneau">caractéristique</a> 0, il existe une belle classification des groupes algébriques unipotents par rapport aux <a href="/w/index.php?title=Alg%C3%A8bre_de_Lie_nilpotente&action=edit&redlink=1" class="new" title="Algèbre de Lie nilpotente (page inexistante)">algèbres de Lie nilpotentes</a>. Rappelons qu'une <a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">algèbre de Lie</a> nilpotente est une sous-algèbre d'une algèbre <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 {gl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {gl}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7074085f669b257cf21f2f443dd433888e2597b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.042ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {gl}}_{n}}"></span> telle que l'action adjointe admette une puissance nulle. En termes de matrices, cela signifie qu'il s'agit d'une sous-algèbre <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> de <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 {n}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">n</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {n}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38f4b73d08583660882d55f903b2f61d30b5fbb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.444ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {n}}_{n}}"></span>, les matrices triangulaires supérieures avec des zéros sur la diagonale : <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_{ij}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a565e93211d1ad5c06a571ff8952ab3dfcff3638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ij}=0}"></span> pour <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 i\leq j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>≤</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\leq j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894ab6e9c9afcfea7d9370399cebe1557bdf9b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i\leq j}"></span> . </p><p>Alors, il y a une <a href="/wiki/%C3%89quivalence_de_cat%C3%A9gories" title="Équivalence de catégories">équivalence de catégories</a> des algèbres de Lie nilpotentes de dimension finie et des groupes algébriques unipotents<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. Cela peut être construit en utilisant la <a href="/wiki/Formule_de_Baker-Campbell-Hausdorff" title="Formule de Baker-Campbell-Hausdorff">série de Baker-Campbell-Hausdorff</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 H(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d87c82092f7817e719251729dc0a55289df0eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.66ex; height:2.843ex;" alt="{\displaystyle H(X,Y)}"></span> : étant donné une algèbre de Lie nilpotente de dimension finie, l'application </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:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}},\quad (X,Y)\mapsto H(X,Y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <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">g</mi> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}},\quad (X,Y)\mapsto H(X,Y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a14db90611aebcad3155ccad2e524ba0901b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.198ex; height:2.843ex;" alt="{\displaystyle H:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}},\quad (X,Y)\mapsto H(X,Y)}"></span></dd></dl> <p>munit <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> d'une structure de groupe algébrique unipotent. </p><p>Dans l'autre sens, l'<a href="/w/index.php?title=Application_exponentielle_(th%C3%A9orie_de_Lie)&action=edit&redlink=1" class="new" title="Application exponentielle (théorie de Lie) (page inexistante)">application exponentielle</a> envoie toute matrice carrée nilpotente sur une matrice unipotente. De plus, si <i>U</i> est un groupe unipotent commutatif, l'application exponentielle induit un <a href="/wiki/Isomorphisme" title="Isomorphisme">isomorphisme</a> de l'algèbre de Lie de <i>U</i> vers <i>U</i> lui-même. </p> <div class="mw-heading mw-heading3"><h3 id="Remarques">Remarques</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=10" title="Modifier la section : Remarques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=10" title="Modifier le code source de la section : Remarques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les groupes unipotents sur un <a href="/wiki/Corps_alg%C3%A9briquement_clos" title="Corps algébriquement clos">corps algébriquement clos</a> de n'importe quelle dimension donnée peuvent en principe être classés, mais en pratique la complexité de la classification augmente très rapidement avec la dimension, de sorte que les gens<sup class="need_ref_tag" style="padding-left:2px;"><a href="/wiki/Aide:Qui" title="Aide:Qui">[Qui ?]</a></sup> ont tendance à abandonner quelque part autour de la dimension 6. </p> <div class="mw-heading mw-heading2"><h2 id="Radical_unipotent">Radical unipotent</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=11" title="Modifier la section : Radical unipotent" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=11" title="Modifier le code source de la section : Radical unipotent"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Le <b>radical unipotent</b> d'un <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">groupe algébrique</a> <i>G</i> est l'ensemble des éléments unipotents du <a href="/w/index.php?title=Radical_d%27un_groupe_alg%C3%A9brique&action=edit&redlink=1" class="new" title="Radical d'un groupe algébrique (page inexistante)">radical</a> de <i>G</i>. C'est un <a href="/wiki/Sous-groupe_normal" title="Sous-groupe normal">sous-groupe normal</a> unipotent connexe de <i>G</i>, et il contient tous les autres sous-groupes de ce type. Un groupe est dit réductif si son radical unipotent est trivial. Si <i>G</i> est réductif alors son radical est un tore. </p> <div class="mw-heading mw-heading2"><h2 id="Décomposition_des_groupes_algébriques"><span id="D.C3.A9composition_des_groupes_alg.C3.A9briques"></span>Décomposition des groupes algébriques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=12" title="Modifier la section : Décomposition des groupes algébriques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=12" title="Modifier le code source de la section : Décomposition des groupes algébriques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les groupes algébriques peuvent être décomposés en groupes unipotents, groupes multiplicatifs et <a href="/wiki/Vari%C3%A9t%C3%A9_ab%C3%A9lienne" title="Variété abélienne">variétés abéliennes</a> mais l'énoncé de la décomposition dépend de la caractéristique de leur <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps</a> de base. </p> <div class="mw-heading mw-heading3"><h3 id="Caractéristique_0"><span id="Caract.C3.A9ristique_0"></span>Caractéristique 0</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=13" title="Modifier la section : Caractéristique 0" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=13" title="Modifier le code source de la section : Caractéristique 0"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En caractéristique 0, il existe un joli théorème de décomposition d'un groupe algébrique <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> reliant sa structure à la structure d'un <a href="/w/index.php?title=Groupe_alg%C3%A9brique_lin%C3%A9aire&action=edit&redlink=1" class="new" title="Groupe algébrique linéaire (page inexistante)">groupe algébrique linéaire</a> et d'une <a href="/wiki/Vari%C3%A9t%C3%A9_ab%C3%A9lienne" title="Variété abélienne">variété abélienne</a>. Il y a une <a href="/wiki/Suite_exacte" title="Suite exacte">suite exacte courte</a> de groupes<sup id="cite_ref-Brion_3-0" class="reference"><a href="#cite_note-Brion-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> </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 0\to M\times U\to G\to A\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <mi>M</mi> <mo>×</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to M\times U\to G\to A\to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e43aa543fabad09fc9158919ff0223d477afbd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:27.416ex; height:2.176ex;" alt="{\displaystyle 0\to M\times U\to G\to A\to 0}"></span></dd></dl> <p>où <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span> est une variété abélienne, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> est de type multiplicatif (c'est-à-dire que <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> est, géométriquement, un produit de tores et de groupes algébriques de la forme <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 _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/267d03f9351dcc8d3d3ac7cad59ea3ba4fecbfef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.62ex; height:2.176ex;" alt="{\displaystyle \mu _{n}}"></span>) et <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> est un groupe unipotent. </p> <div class="mw-heading mw-heading3"><h3 id="Caractéristique_p"><span id="Caract.C3.A9ristique_p"></span>Caractéristique <i>p</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=14" title="Modifier la section : Caractéristique p" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=14" title="Modifier le code source de la section : Caractéristique p"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lorsque la caractéristique du corps de base est <i>p</i>, on peut formuler une description analogue<sup id="cite_ref-Brion_3-1" class="reference"><a href="#cite_note-Brion-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup> pour un groupe algébrique <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> : il existe un sous-groupe minimal <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> tel que </p> <ol><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 G/H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21e7e9d6e3072ec8dd48200d755847154ea5d35c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/H}"></span> est un groupe unipotent ;</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 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> est une extension d'une variété abélienne <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span> par un groupe <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> de type multiplicatif ;</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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> est unique à <a href="/w/index.php?title=Commensurabilit%C3%A9_(th%C3%A9orie_des_groupes)&action=edit&redlink=1" class="new" title="Commensurabilité (théorie des groupes) (page inexistante)">commensurabilité</a> près dans <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> et <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span> est unique à <a href="/w/index.php?title=Isog%C3%A9nie&action=edit&redlink=1" class="new" title="Isogénie (page inexistante)">isogénie</a> près.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Décomposition_de_Jordan"><span id="D.C3.A9composition_de_Jordan"></span>Décomposition de Jordan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=15" title="Modifier la section : Décomposition de Jordan" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=15" title="Modifier le code source de la section : Décomposition de Jordan"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tout élément <i>g</i> d'un groupe algébrique linéaire sur un <a href="/wiki/Corps_parfait" title="Corps parfait">corps parfait</a> peut être écrit de manière unique comme le produit <i>g</i> = <i>g</i><sub><i>u</i></sub>  <i>g</i><sub><i>s</i></sub> d'un élément unipotent <i>g</i><sub><i>u</i></sub> et d'un élément <a href="/wiki/Matrice_semi-simple" title="Matrice semi-simple">semi-simple</a> <i>g</i><sub><i>s</i></sub> qui commutent. Dans le cas du groupe GL<sub><i>n</i></sub>(<b>C</b>), cela signifie essentiellement que toute matrice <a href="/wiki/Nombre_complexe" title="Nombre complexe">complexe</a> inversible est conjuguée au produit d'une matrice diagonale et d'une matrice triangulaire supérieure qui commutent, ce qui est (plus ou moins) la version multiplicative de la <a href="/wiki/D%C3%A9composition_de_Dunford" title="Décomposition de Dunford">décomposition de Jordan–Chevalley</a>. </p><p>Il existe également une version de la décomposition de Jordan pour les groupes : tout groupe algébrique linéaire commutatif sur un corps parfait est le produit d'un groupe unipotent et d'un groupe semi-simple. </p> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=16" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=16" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="plainlinks">« <a class="external text" href="https://en.wikipedia.org/wiki/Unipotent?oldid=1134733182">Unipotent</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Unipotent?action=history">voir la liste des auteurs</a>)</small></span>.</li></ul> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-M252-1"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-M252_1-0">a</a> et <a href="#cite_ref-M252_1-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="Milne"><span class="ouvrage" id="James_S._Milne">James S. <span class="nom_auteur">Milne</span>, <cite style="font-style:normal">« Unipotent algebraic groups »</cite>, dans <cite class="italique">Linear Algebraic Groups</cite> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://www.jmilne.org/math/CourseNotes/iAG200.pdf">lire en ligne</a>)</small>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">252-253</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Linear+Algebraic+Groups&rft.atitle=Unipotent+algebraic+groups&rft.aulast=Milne&rft.aufirst=James+S.&rft.pages=252-253&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Milne"><span class="ouvrage" id="James_S._Milne">James S. <span class="nom_auteur">Milne</span>, <cite style="font-style:normal">« Unipotent algebraic groups »</cite>, dans <cite class="italique">Linear Algebraic Groups</cite> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="https://www.jmilne.org/math/CourseNotes/iAG200.pdf">lire en ligne</a>)</small>, <abbr class="abbr" title="page">p.</abbr> 261<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Linear+Algebraic+Groups&rft.atitle=Unipotent+algebraic+groups&rft.aulast=Milne&rft.aufirst=James+S.&rft.pages=261&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span>.</span> </li> <li id="cite_note-Brion-3"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-Brion_3-0">a</a> et <a href="#cite_ref-Brion_3-1">b</a></sup> </span><span class="reference-text"><span class="ouvrage" id="Brion2018"><span class="ouvrage" id="Michel_Brion2018">Michel <span class="nom_auteur">Brion</span>, « <cite style="font-style:normal">Commutative algebraic groups up to isogeny</cite> », <i>Contemporary Mathematics</i>, <abbr class="abbr" title="volume">vol.</abbr> 705,‎ <time>2018</time>, <abbr class="abbr" title="page">p.</abbr> 8 <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1090/conm/705">10.1090/conm/705</a></span> <span typeof="mw:File"><span title="Inscription nécessaire pour accéder au document"><img alt="Inscription nécessaire" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/14px-Lock-gray-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/18px-Lock-gray-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /></span></span>, <a href="/wiki/ArXiv" title="ArXiv">arXiv</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1602.00222">1602.00222</a></span>, <a href="/wiki/Semantic_Scholar" title="Semantic Scholar">S2CID</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17414231">17414231</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Commutative+algebraic+groups+up+to+isogeny&rft.jtitle=Contemporary+Mathematics&rft.aulast=Brion&rft.aufirst=Michel&rft.date=2018&rft.volume=705&rft.pages=8&rft_id=info%3Adoi%2F10.1090%2Fconm%2F705&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=17" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=17" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Bibliographie">Bibliographie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=18" title="Modifier la section : Bibliographie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=18" title="Modifier le code source de la section : Bibliographie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="Borel1991"><span class="ouvrage" id="Armand_Borel1991"><a href="/wiki/Armand_Borel" title="Armand Borel">Armand <span class="nom_auteur">Borel</span></a>, <cite class="italique">Linear algebraic groups</cite>, New York, Springer, <abbr class="abbr" title="collection">coll.</abbr> « Graduate Texts in Mathematics », <time>1991</time>, <abbr class="abbr" title="deuxième">2<sup>e</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr> (<abbr class="abbr" title="première">1<sup>re</sup></abbr> <abbr class="abbr" title="édition">éd.</abbr> 1969) <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/0-387-97370-2" title="Spécial:Ouvrages de référence/0-387-97370-2"><span class="nowrap">0-387-97370-2</span></a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebraic+groups&rft.place=New+York&rft.pub=Springer&rft.edition=2&rft.aulast=Borel&rft.aufirst=Armand&rft.date=1991&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></li> <li><span class="ouvrage" id="Borel1956"><span class="ouvrage" id="Armand_Borel1956"><a href="/wiki/Armand_Borel" title="Armand Borel">Armand <span class="nom_auteur">Borel</span></a>, « <cite style="font-style:normal">Groupes linéaires algébriques</cite> », <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, second series, <abbr class="abbr" title="volume">vol.</abbr> 64, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 1,‎ <time>1956</time>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">20-82</span> <small style="line-height:1em;">(<a href="/wiki/Digital_Object_Identifier" title="Digital Object Identifier">DOI</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/1969949">10.2307/1969949</a></span>, <a href="/wiki/JSTOR" title="JSTOR">JSTOR</a> <span class="plainlinks noarchive nowrap"><a rel="nofollow" class="external text" href="https://jstor.org/stable/1969949">1969949</a></span>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Groupes+lin%C3%A9aires+alg%C3%A9briques&rft.jtitle=Annals+of+Mathematics&rft.issue=1&rft.aulast=Borel&rft.aufirst=Armand&rft.date=1956&rft.volume=64&rft.pages=20-82&rft_id=info%3Adoi%2F10.2307%2F1969949&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></li> <li><span class="ouvrage" id="Popov2002"><span class="ouvrage" id="Vladimir_L._Popov2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Vladimir_L._Popov_(math%C3%A9maticien)" title="Vladimir L. Popov (mathématicien)">Vladimir L. <span class="nom_auteur">Popov</span></a>, <cite style="font-style:normal" lang="en">« Unipotent element »</cite>, dans <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a>, <cite class="italique" lang="en"><a href="/wiki/Encyclop%C3%A6dia_of_Mathematics" title="Encyclopædia of Mathematics">Encyclopædia of Mathematics</a></cite>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <time>2002</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1556080104" title="Spécial:Ouvrages de référence/978-1556080104"><span class="nowrap">978-1556080104</span></a>, <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=Unipotent_element">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Encyclop%C3%A6dia+of+Mathematics&rft.atitle=Unipotent+element&rft.pub=Springer&rft.aulast=Popov&rft.aufirst=Vladimir+L.&rft.date=2002&rft.isbn=978-1556080104&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></li> <li><span class="ouvrage" id="Popov2002"><span class="ouvrage" id="Vladimir_L._Popov2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Vladimir_L._Popov_(math%C3%A9maticien)" title="Vladimir L. Popov (mathématicien)">Vladimir L. <span class="nom_auteur">Popov</span></a>, <cite style="font-style:normal" lang="en">« Unipotent group »</cite>, dans <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a>, <cite class="italique" lang="en"><a href="/wiki/Encyclop%C3%A6dia_of_Mathematics" title="Encyclopædia of Mathematics">Encyclopædia of Mathematics</a></cite>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <time>2002</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1556080104" title="Spécial:Ouvrages de référence/978-1556080104"><span class="nowrap">978-1556080104</span></a>, <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=Unipotent_group">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Encyclop%C3%A6dia+of+Mathematics&rft.atitle=Unipotent+group&rft.pub=Springer&rft.aulast=Popov&rft.aufirst=Vladimir+L.&rft.date=2002&rft.isbn=978-1556080104&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></li> <li><span class="ouvrage" id="Suprunenko2002"><span class="ouvrage" id="Dmitrii_A._Suprunenko2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Dmitrii A. <span class="nom_auteur">Suprunenko</span>, <cite style="font-style:normal" lang="en">« Unipotent matrix »</cite>, dans <a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Michiel Hazewinkel</a>, <cite class="italique" lang="en"><a href="/wiki/Encyclop%C3%A6dia_of_Mathematics" title="Encyclopædia of Mathematics">Encyclopædia of Mathematics</a></cite>, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <time>2002</time> <small style="line-height:1em;">(<a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Sp%C3%A9cial:Ouvrages_de_r%C3%A9f%C3%A9rence/978-1556080104" title="Spécial:Ouvrages de référence/978-1556080104"><span class="nowrap">978-1556080104</span></a>, <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=Unipotent_matrix">lire en ligne</a>)</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Encyclop%C3%A6dia+of+Mathematics&rft.atitle=Unipotent+matrix&rft.pub=Springer&rft.aulast=Suprunenko&rft.aufirst=Dmitrii+A.&rft.date=2002&rft.isbn=978-1556080104&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AUnipotent"></span></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unipotent&veaction=edit&section=19" title="Modifier la section : Articles connexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Unipotent&action=edit&section=19" title="Modifier le code source de la section : Articles connexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Groupe_r%C3%A9ductif" title="Groupe réductif">Groupe réductif</a></li> <li><a href="/w/index.php?title=Repr%C3%A9sentation_unipotente&action=edit&redlink=1" class="new" title="Représentation unipotente (page inexistante)">Représentation unipotente</a> <a href="https://en.wikipedia.org/wiki/Unipotent_representation" class="extiw" title="en:Unipotent representation"><span class="indicateur-langue" title="Article en anglais : « Unipotent representation »">(en)</span></a></li> <li><a href="/w/index.php?title=Th%C3%A9orie_de_Deligne-Lusztig&action=edit&redlink=1" class="new" title="Théorie de Deligne-Lusztig (page inexistante)">Théorie de Deligne-Lusztig</a> <a href="https://en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig_theory" class="extiw" title="en:Deligne–Lusztig theory"><span class="indicateur-langue" title="Article en anglais : « Deligne–Lusztig theory »">(en)</span></a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" 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