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A representation of a connected algebraic reductive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176901.png" /> as the union of double cosets of a [[Borel subgroup|Borel subgroup]], parametrized by the [[Weyl group|Weyl group]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176902.png" />. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176903.png" /> be opposite Borel subgroups of a reductive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176904.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176905.png" /> be the respective unipotent parts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176906.png" /> (cf. [[Linear algebraic group|Linear algebraic group]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176907.png" /> be the Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176908.png" />. In what follows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b0176909.png" /> denotes both an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769010.png" /> and its representative in the normalizer of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769011.png" />, since the construction presented below is independent of the representative chosen. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769012.png" /> will then be considered for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769013.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769014.png" /> is then representable as the union of the non-intersecting double cosets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769015.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769016.png" />), and the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769017.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769018.png" /> is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769019.png" />. Namely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769020.png" /> is a fixed (with respect to the left shifts by elements from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769021.png" />) point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769022.png" /> (such a point always exists, cf. [[Borel fixed-point theorem|Borel fixed-point theorem]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769023.png" /> will be the union of non-intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769024.png" />-orbits of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769026.png" /> (cf. [[Algebraic group of transformations|Algebraic group of transformations]]), and the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769027.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769028.png" /> is an isomorphism of algebraic varieties. All groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769029.png" />, being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769030.png" />-orbits is a cell in the sense of algebraic topology so that the homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769031.png" /> can be calculated. The existence of a Bruhat decomposition for a number of classical groups was established in 1956 by F. Bruhat, and was proved in the general case by C. Chevalley [[#References|[3]]]. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769033.png" />-points of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769034.png" />-defined algebraic group [[#References|[2]]], the role of Borel subgroups being played by minimal parabolic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769035.png" />-subgroups, the role of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769036.png" /> by their unipotent radicals; the Weyl <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769037.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769038.png" /> or the relative Weyl group was considered instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017690/b01769039.png" />.
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{{MSC|20G}}
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{{TEX|done}}
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The ''Bruhat decomposition'' is
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a representation of a connected split algebraic reductive group $G$, as the union of double cosets of a
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[[Borel subgroup|Borel subgroup]], parametrized by the
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[[Weyl group|Weyl group]] of $G$. More exactly, let $B,B^-$ be opposite Borel subgroups of a reductive group $G$; let $U,U^-$ be the respective unipotent parts of $B,B^-$ (cf.
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[[Linear algebraic group|Linear algebraic group]]) and let $W$ be the Weyl group of $G$. In what follows $w$ denotes both an element of $W$ and its representative in the normalizer of the torus $B\cap B^-$, since the construction presented below is independent of the representative chosen. The group $U_w^- = U\cap wU^-w^{-1}$ will then be considered for each $w\in W$. The group $G$ is then representable as the union of the non-intersecting double cosets $BwB$ ($w\in W$), and the morphism $U_w^-\times B \to BwB$  defined by $(x,y)\mapsto xwy$ is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety $G/B$. Namely, if $x_0$ is a fixed (with respect to the left shifts by elements from $B$) point of $G/B$ (such a point always exists, cf.
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[[Borel fixed-point theorem|Borel fixed-point theorem]]), $G/B$ will be the union of non-intersecting $U$-orbits of the type $U(w(x_0))$, $w\in W$ (cf.
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[[Algebraic group of transformations|Algebraic group of transformations]]), and the morphism $U_w^- \to Uw(x_0)$ ($u\mapsto u(w(x_0))$) is an isomorphism of algebraic varieties. All groups $U_w^-$, being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above $U$-orbits is a cell in the sense of algebraic topology so that the homology of $G/B$ can be calculated. The existence of a Bruhat decomposition for a number of classical groups was established in 1956 by F. Bruhat, and was proved in the general case by C. Chevalley
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{{Cite|Ch}}. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the non split groups $G_k$ of $k$-points of a $k$-defined algebraic group
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{{Cite|BoTi}}, the role of Borel subgroups being played by minimal parabolic $k$-subgroups, the role of the groups $U$ by their unipotent radicals; the Weyl $k$-group $W_k$ or the relative Weyl group was considered instead of $W$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel,  "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </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  {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C. Chevalley,  "Classification des groupes de Lie algébriques" , '''2''' , Ecole Norm. Sup.  (1958)  {{MR|}} {{ZBL|0092.26301}} </TD></TR></table>
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{|
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|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel,  "Linear algebraic groups", Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}}
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|-
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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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|-
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|valign="top"|{{Ref|Ch}}||valign="top"| Séminaire C. Chevalley,  "Classification des groupes de Lie algébriques", '''2''', Ecole Norm. Sup.  (1956-1958)  {{MR|0106966}} {{ZBL|0092.26301}}
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|-
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Latest revision as of 14:13, 27 April 2012

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

The Bruhat decomposition is a representation of a connected split algebraic reductive group $G$, as the union of double cosets of a Borel subgroup, parametrized by the Weyl group of $G$. More exactly, let $B,B^-$ be opposite Borel subgroups of a reductive group $G$; let $U,U^-$ be the respective unipotent parts of $B,B^-$ (cf. Linear algebraic group) and let $W$ be the Weyl group of $G$. In what follows $w$ denotes both an element of $W$ and its representative in the normalizer of the torus $B\cap B^-$, since the construction presented below is independent of the representative chosen. The group $U_w^- = U\cap wU^-w^{-1}$ will then be considered for each $w\in W$. The group $G$ is then representable as the union of the non-intersecting double cosets $BwB$ ($w\in W$), and the morphism $U_w^-\times B \to BwB$ defined by $(x,y)\mapsto xwy$ is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety $G/B$. Namely, if $x_0$ is a fixed (with respect to the left shifts by elements from $B$) point of $G/B$ (such a point always exists, cf. Borel fixed-point theorem), $G/B$ will be the union of non-intersecting $U$-orbits of the type $U(w(x_0))$, $w\in W$ (cf. Algebraic group of transformations), and the morphism $U_w^- \to Uw(x_0)$ ($u\mapsto u(w(x_0))$) is an isomorphism of algebraic varieties. All groups $U_w^-$, being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above $U$-orbits is a cell in the sense of algebraic topology so that the homology of $G/B$ can be calculated. The existence of a Bruhat decomposition for a number of classical groups was established in 1956 by F. Bruhat, and was proved in the general case by C. Chevalley [Ch]. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the non split groups $G_k$ of $k$-points of a $k$-defined algebraic group [BoTi], the role of Borel subgroups being played by minimal parabolic $k$-subgroups, the role of the groups $U$ by their unipotent radicals; the Weyl $k$-group $W_k$ or the relative Weyl group was considered instead of $W$.

References

[Bo] A. Borel, "Linear algebraic groups", Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[BoTi] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[Ch] Séminaire C. Chevalley, "Classification des groupes de Lie algébriques", 2, Ecole Norm. Sup. (1956-1958) MR0106966 Zbl 0092.26301
How to Cite This Entry:
Bruhat decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bruhat_decomposition&oldid=21819
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article