Bruhat decomposition
A representation of a connected algebraic reductive group as the union of double cosets of a Borel subgroup, parametrized by the Weyl group of
. More exactly, let
be opposite Borel subgroups of a reductive group
; let
be the respective unipotent parts of
(cf. Linear algebraic group) and let
be the Weyl group of
. In what follows
denotes both an element of
and its representative in the normalizer of the torus
, since the construction presented below is independent of the representative chosen. The group
will then be considered for each
. The group
is then representable as the union of the non-intersecting double cosets
(
), and the morphism
is an isomorphism of algebraic varieties. An even more precise formulation of the Bruhat decomposition will yield a cellular decomposition of the projective variety
. Namely, if
is a fixed (with respect to the left shifts by elements from
) point of
(such a point always exists, cf. Borel fixed-point theorem),
will be the union of non-intersecting
-orbits of the type
,
(cf. Algebraic group of transformations), and the morphism
is an isomorphism of algebraic varieties. All groups
, being varieties, are isomorphic to an affine space; if the ground field is the field of complex numbers, then each of the above
-orbits is a cell in the sense of algebraic topology so that the homology of
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 [3]. A. Borel and J. Tits generalized the construction of Bruhat decompositions to the groups
of
-points of a
-defined algebraic group [2], the role of Borel subgroups being played by minimal parabolic
-subgroups, the role of the groups
by their unipotent radicals; the Weyl
-group
or the relative Weyl group was considered instead of
.
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[2] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 |
[3] | C. Chevalley, "Classification des groupes de Lie algébriques" , 2 , Ecole Norm. Sup. (1958) |
Bruhat decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bruhat_decomposition&oldid=17813