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A maximal connected solvable algebraic subgroup of a [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171601.png" />. Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171602.png" />. A. Borel [[#References|[1]]] was the first to carry out a systematic study of maximal connected solvable subgroups of algebraic groups. Borel subgroups can be characterized as minimal parabolic subgroups, i.e. algebraic subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171603.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171604.png" /> for which the quotient variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171605.png" /> is projective. All Borel subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171606.png" /> are conjugate and, if the Borel subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171608.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b0171609.png" /> are defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716012.png" /> are conjugate by an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716013.png" />. The intersection of any two Borel subgroups of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716014.png" /> contains a maximal torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716015.png" />; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716016.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716017.png" /> is a [[Reductive group|reductive group]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716018.png" /> is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716019.png" />. In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716020.png" />. Nevertheless, maximal solvable subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716021.png" /> which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716022.png" /> coincides with its unipotent part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716023.png" />, while the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716024.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716025.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716026.png" />. If the characteristic of the ground field is 0, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716027.png" /> is the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716028.png" />, then the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716030.png" /> which is the Lie algebra of the Borel subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716032.png" /> is often referred to as a Borel subalgebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716033.png" />. The Borel subalgebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716034.png" /> are its maximal solvable subalgebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716035.png" /> is defined over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716036.png" />, the parabolic subgroups which are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716037.png" /> and are minimal for this property, play a role in the theory of algebraic groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716038.png" /> similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017160/b01716039.png" /> [[#References|[2]]].
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A maximal connected solvable algebraic subgroup of a
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[[Linear algebraic group|linear algebraic group]] $G$. Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group $\textrm{GL}(n)$. A. Borel
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{{Cite|Bo}} was the first to carry out a systematic study of maximal connected [[solvable group | solvable]] subgroups of algebraic groups. Borel subgroups can be characterized as minimal [[parabolic subgroup | parabolic subgroups]], i.e. algebraic subgroups $H$ of the group $G$ for which the quotient variety $G/H$ is projective. All Borel subgroups of $G$ are conjugate and, if the Borel subgroups $B_1$, $B_2$ and the group $G$ are defined over a field $k$, $B_1$ and $B_2$ are conjugate by an element of $G(k)$. The intersection of any two Borel subgroups of a group $G$ contains a maximal torus of $G$; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in $G$ if and only if $G$ is a
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[[Reductive group|reductive group]]. If $G$ is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in $G$. In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of $G$. Nevertheless, maximal solvable subgroups in $G$ which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup $B$ coincides with its unipotent part $B_u$, while the normalizer of $B_u$ in $G$ coincides with $B$. If the characteristic of the ground field is 0, and $\def\fg{\mathfrak{g}}\fg$ is the Lie algebra of $G$, then the subalgebra $\mathfrak{b}$ of $\fg$ which is the Lie algebra of the Borel subgroup $B$ of $G$ is often referred to as a Borel subalgebra in $\fg$. The Borel subalgebras in $\fg$ are its maximal solvable subalgebras. If $G$ is defined over an arbitrary field $k$, the parabolic subgroups which are defined over $k$ and are minimal for this property, play a role in the theory of algebraic groups over $k$ similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element of $G(k)$
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{{Cite|BoTi}}.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel,  "Groupes linéaires algébriques"  ''Ann. of Math. (2)'' , '''64''' :  1  (1956)  pp. 20–82</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel,  J. Tits,  "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150</TD></TR></table>
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|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel,  "Groupes linéaires algébriques"  ''Ann. of Math. (2)'', '''64''' :  1  (1956)  pp. 20–82 {{MR|0093006}} {{ZBL|0070.26104}}
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|valign="top"|{{Ref|BoTi}}||valign="top"| A. Borel,  J. Tits,  "Groupes réductifs"  ''Publ. Math. IHES'', '''27'''  (1965)  pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}}
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Latest revision as of 17:51, 27 April 2012

2020 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

A maximal connected solvable algebraic subgroup of a linear algebraic group $G$. Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group $\textrm{GL}(n)$. A. Borel [Bo] was the first to carry out a systematic study of maximal connected solvable subgroups of algebraic groups. Borel subgroups can be characterized as minimal parabolic subgroups, i.e. algebraic subgroups $H$ of the group $G$ for which the quotient variety $G/H$ is projective. All Borel subgroups of $G$ are conjugate and, if the Borel subgroups $B_1$, $B_2$ and the group $G$ are defined over a field $k$, $B_1$ and $B_2$ are conjugate by an element of $G(k)$. The intersection of any two Borel subgroups of a group $G$ contains a maximal torus of $G$; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in $G$ if and only if $G$ is a reductive group. If $G$ is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in $G$. In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of $G$. Nevertheless, maximal solvable subgroups in $G$ which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup $B$ coincides with its unipotent part $B_u$, while the normalizer of $B_u$ in $G$ coincides with $B$. If the characteristic of the ground field is 0, and $\def\fg{\mathfrak{g}}\fg$ is the Lie algebra of $G$, then the subalgebra $\mathfrak{b}$ of $\fg$ which is the Lie algebra of the Borel subgroup $B$ of $G$ is often referred to as a Borel subalgebra in $\fg$. The Borel subalgebras in $\fg$ are its maximal solvable subalgebras. If $G$ is defined over an arbitrary field $k$, the parabolic subgroups which are defined over $k$ and are minimal for this property, play a role in the theory of algebraic groups over $k$ similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element of $G(k)$ [BoTi].

References

[Bo] A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2), 64 : 1 (1956) pp. 20–82 MR0093006 Zbl 0070.26104
[BoTi] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
How to Cite This Entry:
Borel subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_subgroup&oldid=14476
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article