Lie Groups pp 243-256 | Cite as

The Bruhat Decomposition

  • Daniel Bump
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)


The Bruhat decomposition was discovered quite late in the history of Lie groups, which is surprising in view of its fundamental importance. It was preceded by Ehresmann’s discovery of a closely related cell decomposition for flag manifolds. The Bruhat decomposition was axiomatized by Tits in the notion of a Group with (B, N) pair or Tits’ system. This is a generalization of the notion of a Coxeter group, and indeed every (B, N) gives rise to a Coxeter group. We have remarked after Theorem 25.1 that Coxeter groups always act on simplicial complexes whose geometry is closely connected with their properties. As it turns out a group with (B N) pair also acts on a simplicial complex, the Tits’ building. We will not have space to discuss this important concept but see Tits [163] and Abramenko and Brown [1].


Line Bundle Simplicial Complex Weyl Group Parabolic Subgroup Maximal Torus 
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  1. 1.
    Peter Abramenko and Kenneth S. Brown. Buildings, volume 248 of Graduate Texts in Mathematics. Springer, New York, 2008. Theory and applications.Google Scholar
  2. 23.
    Nicolas Bourbaki. Lie groups and Lie algebras. Chapters  4 6. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley.
  3. 60.
    R. Gunning and H. Rossi. Analytic functions of several complex variables. Prentice-Hall Inc., Englewood Cliffs, N.J., 1965.Google Scholar
  4. 163.
    Jacques Tits. Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin, 1974.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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