View source for Bruhat decomposition

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The ''Bruhat decomposition'' is 
a representation of a connected split algebraic reductive group $G$, as the union of double cosets of a
[[Borel subgroup|Borel subgroup]], parametrized by the
[[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.
[[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.
[[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.
[[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
{{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
{{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====
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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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|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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|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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