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Difference between revisions of "Bruhat decomposition"

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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)</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><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,  "Classification des groupes de Lie algébriques" , '''2''' , Ecole Norm. Sup.  (1958)</TD></TR></table>
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<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>

Revision as of 10:02, 24 March 2012

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) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[3] C. Chevalley, "Classification des groupes de Lie algébriques" , 2 , Ecole Norm. Sup. (1958) 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