Sous-groupe de Borel

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Disponible en 4 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-4" 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">4 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/Parabolische_Untergruppe" title="Parabolische Untergruppe – allemand" lang="de" hreflang="de" data-title="Parabolische Untergruppe" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Borel_subgroup" title="Borel subgroup – anglais" lang="en" hreflang="en" data-title="Borel subgroup" 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%B3%B4%EB%A0%90_%EB%B6%80%EB%B6%84%EA%B5%B0" title="보렐 부분군 – coréen" lang="ko" hreflang="ko" data-title="보렐 부분군" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%91%D0%BE%D1%80%D0%B5%D0%BB%D1%8F" title="Подгруппа Бореля – russe" lang="ru" hreflang="ru" data-title="Подгруппа Бореля" data-language-autonym="Русский" data-language-local-name="russe" 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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>Dans la théorie des <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">groupes algébriques</a>, un <b>sous-groupe de Borel</b> d'un <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">groupe algébrique</a> <i>G</i> est un <a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">sous-groupe algébrique</a> <a href="/wiki/Groupe_r%C3%A9soluble" title="Groupe résoluble">résoluble</a>, <a href="/wiki/Topologie_de_Zariski" title="Topologie de Zariski">fermé</a>, <a href="/wiki/Connexit%C3%A9_(math%C3%A9matiques)" title="Connexité (mathématiques)">connexe</a> et maximal pour ces propriétés. Par exemple, dans le <a href="/wiki/Groupe_g%C3%A9n%C3%A9ral_lin%C3%A9aire" title="Groupe général linéaire">groupe général linéaire</a> GL<sub><i>n</i></sub> (matrices inversibles <i>n</i>×<i>n</i>), le sous-groupe des <a href="/wiki/Matrice_triangulaire" title="Matrice triangulaire">matrices triangulaires supérieures</a> inversibles est un sous-groupe de Borel. </p><p>Pour les groupes réalisés sur des <a href="/wiki/Corps_alg%C3%A9briquement_clos" title="Corps algébriquement clos">corps algébriquement clos</a>, il existe une seule <a href="/wiki/Action_par_conjugaison" title="Action par conjugaison">classe de conjugaison</a> de sous-groupes de Borel. </p><p>Les sous-groupes de Borel sont l'un des deux ingrédients clés pour comprendre la structure des groupes algébriques simples (ou plus généralement <a href="/wiki/Groupe_r%C3%A9ductif" title="Groupe réductif">réductifs</a>), dans la théorie développée par <a href="/wiki/Jacques_Tits" title="Jacques Tits">Jacques Tits</a> pour les groupes munis d'une <i><a href="/wiki/BN-paire" title="BN-paire">BN-paire</a></i> (ou <i>système de Tits</i>). Ici le groupe <i>B</i> est un sous-groupe de Borel et <i>N</i> est le normalisateur d'un <a href="/wiki/Tore_maximal" title="Tore maximal">tore maximal</a> contenu dans <i>B</i>. </p><p>Cette notion a été introduite par <a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a>, qui a joué un rôle de premier plan dans le développement de la théorie des groupes algébriques. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Sous-groupes_paraboliques">Sous-groupes paraboliques</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=1" title="Modifier la section : Sous-groupes paraboliques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sous-groupe_de_Borel&action=edit&section=1" title="Modifier le code source de la section : Sous-groupes paraboliques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Les sous-groupes emboîtés entre un sous-groupe de Borel <i>B</i> et le groupe ambiant <i>G</i> sont appelés <b>sous-groupes paraboliques</b>. Les sous-groupes paraboliques <i>P</i> sont également caractérisés, parmi les sous-groupes algébriques, par la condition que <i>G</i>/<i>P</i> est une <a href="/wiki/Vari%C3%A9t%C3%A9_compl%C3%A8te" title="Variété complète">variété complète</a>. Si l'on travaille sur des corps algébriquement clos, les sous-groupes de Borel s'avèrent être les <b>sous-groupes paraboliques minimaux</b> dans ce sens. Autrement dit, <i>B</i> est un sous-groupe de Borel lorsque l'espace homogène <i>G/B</i> est une variété complète « aussi grande que possible ». </p><p>Pour un groupe algébrique simple <i>G</i>, l'ensemble des <a href="/wiki/Action_par_conjugaison" title="Action par conjugaison">classes de conjugaison</a> des sous-groupes paraboliques est en bijection avec les parties de l'ensemble des sommets du <a href="/wiki/Syst%C3%A8me_de_racines" title="Système de racines">diagramme de Dynkin</a> correspondant ; le sous-groupe de Borel correspond à l'ensemble vide et <i>G</i> lui-même correspond à l'ensemble de tous les sommets. (De façon générale, chaque sommet du diagramme de Dynkin détermine une racine négative simple et donc un « sous-groupe radiciel » de dimension 1 de <i>G</i>. De la sorte, un sous-ensemble des sommets détermine un sous-groupe parabolique, engendré par <i>B</i> et les sous-groupes radiciels négatifs correspondants. De plus, tout sous-groupe parabolique est conjugué à un tel sous-groupe parabolique.) </p> <div class="mw-heading mw-heading2"><h2 id="Exemple">Exemple</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=2" title="Modifier la section : Exemple" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sous-groupe_de_Borel&action=edit&section=2" title="Modifier le code source de la section : Exemple"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div><p> 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 G=\operatorname {GL} _{4}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\operatorname {GL} _{4}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a12ef0d8d8b230c3a48e4d94faf905346f929a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.744ex; height:2.843ex;" alt="{\displaystyle G=\operatorname {GL} _{4}(\mathbb {C} )}"></span>. Un sous-groupe Borel <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></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 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> est l'ensemble des matrices triangulaires supérieures</p><blockquote><p><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 \left\{A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}:\det(A)\neq 0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>:</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}:\det(A)\neq 0\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/198292bef49877966d454dc3461007dc6be2a0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:45.198ex; height:12.509ex;" alt="{\displaystyle \left\{A={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}:\det(A)\neq 0\right\}}"></span></p></blockquote><p>et les sous-groupes paraboliques propres maximaux 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 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> contenant <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> sont</p><blockquote><p><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 \left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&a_{32}&a_{33}&a_{34}\\0&a_{42}&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>43</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&a_{32}&a_{33}&a_{34}\\0&a_{42}&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a79d4d23a29325a25c6771b967282c17906523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:87.939ex; height:12.509ex;" alt="{\displaystyle \left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\0&a_{22}&a_{23}&a_{24}\\0&a_{32}&a_{33}&a_{34}\\0&a_{42}&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\0&0&a_{33}&a_{34}\\0&0&a_{43}&a_{44}\end{bmatrix}}\right\},{\text{ }}\left\{{\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\0&0&0&a_{44}\end{bmatrix}}\right\}.}"></span></p></blockquote><p>Par ailleurs, un tore maximal 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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> est</p><blockquote><p><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 \left\{{\begin{bmatrix}a_{11}&0&0&0\\0&a_{22}&0&0\\0&0&a_{33}&0\\0&0&0&a_{44}\end{bmatrix}}:a_{11}\cdot a_{22}\cdot a_{33}\cdot a_{44}\neq 0\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>:</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>44</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{bmatrix}a_{11}&0&0&0\\0&a_{22}&0&0\\0&0&a_{33}&0\\0&0&0&a_{44}\end{bmatrix}}:a_{11}\cdot a_{22}\cdot a_{33}\cdot a_{44}\neq 0\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ef60a9e4f22dfe36e4f94226f929a8b17e5fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:52.069ex; height:12.509ex;" alt="{\displaystyle \left\{{\begin{bmatrix}a_{11}&0&0&0\\0&a_{22}&0&0\\0&0&a_{33}&0\\0&0&0&a_{44}\end{bmatrix}}:a_{11}\cdot a_{22}\cdot a_{33}\cdot a_{44}\neq 0\right\}.}"></span></p></blockquote><p>Ce tore est isomorphe au tore 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 (\mathbb {C} ^{*})^{4}={\text{Spec}}(\mathbb {C} [x^{\pm 1},y^{\pm 1},z^{\pm 1},w^{\pm 1}])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>±</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {C} ^{*})^{4}={\text{Spec}}(\mathbb {C} [x^{\pm 1},y^{\pm 1},z^{\pm 1},w^{\pm 1}])}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/692f93838084cd1eb1bb9aba6d62670e3a10117c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.803ex; height:3.176ex;" alt="{\displaystyle (\mathbb {C} ^{*})^{4}={\text{Spec}}(\mathbb {C} [x^{\pm 1},y^{\pm 1},z^{\pm 1},w^{\pm 1}])}"></span><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. </p><div class="mw-heading mw-heading2"><h2 id="Algèbre_de_Lie"><span id="Alg.C3.A8bre_de_Lie"></span>Algèbre de Lie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=3" title="Modifier la section : Algèbre de Lie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sous-groupe_de_Borel&action=edit&section=3" title="Modifier le code source de la section : Algèbre de Lie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pour le cas particulier d'une <a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">algèbre de Lie</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 {\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> avec une <a href="/wiki/Sous-alg%C3%A8bre_de_Cartan" title="Sous-algèbre de Cartan">sous-algèbre de Cartan</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 {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"></span>, étant donné un <a href="/wiki/Relation_d%27ordre" title="Relation d'ordre">ordre</a> 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 {\mathfrak {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"></span>, la <a href="/w/index.php?title=Sous-alg%C3%A8bre_de_Borel&action=edit&redlink=1" class="new" title="Sous-algèbre de Borel (page inexistante)">sous-algèbre de Borel</a> correspondante est la somme directe 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 {h}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">h</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {h}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f80a9d9b4cf9b0b6f562d5eff0f290da478ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.211ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {h}}}"></span> et des <a href="/wiki/Poids_(th%C3%A9orie_des_repr%C3%A9sentations)" title="Poids (théorie des représentations)">espaces de poids</a> 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 {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> ayant un poids positif. Une sous-algèbre de Lie 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 {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> contenant une sous-algèbre de Borel est appelée une <a href="/w/index.php?title=Alg%C3%A8bre_de_Lie_parabolique&action=edit&redlink=1" class="new" title="Algèbre de Lie parabolique (page inexistante)">algèbre de Lie parabolique</a>. </p> <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=Sous-groupe_de_Borel&veaction=edit&section=4" 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=Sous-groupe_de_Borel&action=edit&section=4" 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="Articles_connexes">Articles connexes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=5" 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=Sous-groupe_de_Borel&action=edit&section=5" 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_hyperbolique" title="Groupe hyperbolique">Groupe hyperbolique</a></li> <li><a href="/wiki/Sous-groupe_de_Cartan" title="Sous-groupe de Cartan">Sous-groupe de Cartan</a></li> <li><a href="/w/index.php?title=Sous-groupe_mirabolique&action=edit&redlink=1" class="new" title="Sous-groupe mirabolique (page inexistante)">Sous-groupe mirabolique</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=6" 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=Sous-groupe_de_Borel&action=edit&section=6" 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/Borel_subgroup?oldid=1138173188">Borel subgroup</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Borel_subgroup?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-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Brion2003"><span class="ouvrage" id="Michel_Brion2003"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Michel <span class="nom_auteur">Brion</span>, « <a rel="nofollow" class="external text" href="https://www-fourier.ujf-grenoble.fr/~mbrion/lecturesrev.pdf"><cite style="font-style:normal;" lang="en">Lectures on the geometry of flag varieties</cite></a> », <time>2003</time></span></span> : notes de l'école d'été "Schubert Varieties" (Varsovie), 59 pages.</span> </li> </ol></div> </div> <ul><li><span class="ouvrage" id="Seitz1995"><span class="ouvrage" id="Gary_Seitz1995">Gary <span class="nom_auteur">Seitz</span>, <cite style="font-style:normal">« Algebraic Groups »</cite>, dans B. Hartley, G. M. Seitz, A. V. Borovik et R. M. Bryant, <cite class="italique">Finite and Locally Finite Groups</cite>, <abbr class="abbr" title="volume">vol.</abbr> 471, <abbr class="abbr" title="collection">coll.</abbr> « Nato Science Series C », <time>1995</time>, xii+458 <abbr class="abbr" title="pages">p.</abbr> <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/978-0-7923-3669-3" title="Spécial:Ouvrages de référence/978-0-7923-3669-3"><span class="nowrap">978-0-7923-3669-3</span></a>, <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/10.1007/978-94-011-0329-9">10.1007/978-94-011-0329-9</a></span>)</small>, <abbr class="abbr" title="pages">p.</abbr> <span class="nowrap">45-70</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Finite+and+Locally+Finite+Groups&rft.atitle=Algebraic+Groups&rft.aulast=Seitz&rft.aufirst=Gary&rft.date=1995&rft.volume=471&rft.pages=45-70&rft.tpages=xii%2B458+p.&rft_id=info%3Adoi%2F10.1007%2F978-94-011-0329-9&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASous-groupe+de+Borel"></span></span></span></li> <li><span class="ouvrage" id="Humphreys1972"><span class="ouvrage" id="James_E._Humphreys1972"><a href="/wiki/James_E._Humphreys" title="James E. Humphreys">James E. <span class="nom_auteur">Humphreys</span></a>, <cite class="italique">Linear Algebraic Groups</cite>, <abbr class="abbr" title="volume">vol.</abbr> 21, New York, Springer, <abbr class="abbr" title="collection">coll.</abbr> « Graduate Texts in Mathematics », <time>1972</time>, xvi+248 <abbr class="abbr" title="pages">p.</abbr> <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/0-387-90108-6" title="Spécial:Ouvrages de référence/0-387-90108-6"><span class="nowrap">0-387-90108-6</span></a>, <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/10.1007/978-1-4684-9443-3">10.1007/978-1-4684-9443-3</a></span>)</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.aulast=Humphreys&rft.aufirst=James+E.&rft.date=1972&rft.volume=21&rft.tpages=xvi%2B248+p.&rft.isbn=0-387-90108-6&rft_id=info%3Adoi%2F10.1007%2F978-1-4684-9443-3&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASous-groupe+de+Borel"></span></span></span></li> <li><span class="ouvrage" id="Borel2001"><span class="ouvrage" id="Armand_Borel2001"><a href="/wiki/Armand_Borel" title="Armand Borel">Armand <span class="nom_auteur">Borel</span></a>, <cite class="italique">Essays in the History of Lie Groups and Algebraic Groups</cite>, <abbr class="abbr" title="volume">vol.</abbr> 21, Providence, RI, American Mathematical Society et London Mathematical Society, <abbr class="abbr" title="collection">coll.</abbr> « History of Mathematics », <time>2001</time>, xiii+184 <abbr class="abbr" title="pages">p.</abbr> <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/0-8218-0288-7" title="Spécial:Ouvrages de référence/0-8218-0288-7"><span class="nowrap">0-8218-0288-7</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=Essays+in+the+History+of+Lie+Groups+and+Algebraic+Groups&rft.place=Providence%2C+RI&rft.pub=American+Mathematical+Society+et+London+Mathematical+Society&rft.aulast=Borel&rft.aufirst=Armand&rft.date=2001&rft.volume=21&rft.tpages=xiii%2B184+p.&rft.isbn=0-8218-0288-7&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASous-groupe+de+Borel"></span></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sous-groupe_de_Borel&veaction=edit&section=7" title="Modifier la section : Liens externes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sous-groupe_de_Borel&action=edit&section=7" title="Modifier le code source de la section : Liens externes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="ouvrage" id="2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <cite style="font-style:normal" lang="en">« Borel subgroup »</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=Borel_subgroup">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=Borel+subgroup&rft.pub=Springer&rft.date=2002&rft.isbn=978-1556080104&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASous-groupe+de+Borel"></span></span></li> <li><span class="ouvrage" id="2002"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <cite style="font-style:normal" lang="en">« Parabolic subgroup »</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=Parabolic_subgroup">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=Parabolic+subgroup&rft.pub=Springer&rft.date=2002&rft.isbn=978-1556080104&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ASous-groupe+de+Borel"></span></span></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail de l’algèbre"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/24px-Arithmetic_symbols.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/36px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/48px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail:Algèbre">Portail de l’algèbre</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Ce document provient de « <a dir="ltr" href="https://fr.wikipedia.org/w/index.php?title=Sous-groupe_de_Borel&oldid=209050095">https://fr.wikipedia.org/w/index.php?title=Sous-groupe_de_Borel&oldid=209050095</a> ».</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Cat%C3%A9gorie:Accueil" title="Catégorie:Accueil">Catégorie</a> : <ul><li><a href="/wiki/Cat%C3%A9gorie:Groupe_alg%C3%A9brique" title="Catégorie:Groupe algébrique">Groupe algébrique</a></li></ul></div><div 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