Lie Groups pp 86-93 | Cite as


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


A complex manifold M is constructed analogously to a smooth manifold. We specify an atlas U = {(U, ø)}, where each chart UM is an open set and ø: U → ℂm is a homeomorphism of U onto its image that is assumed to be open in ℂm. It is assumed that the transition functions ψ o ø −1 : ø(UV) → ψ (UV) are holomorphic for any two charts (U, ø) and (V,ψ). A complex Lie group (or complex analytic group) is a Hausdorff topological group that is a complex manifold in which the multiplication and inversion maps G×GG and GG are holomorphic. The Lie algebra of a complex Lie group is a complex Lie algebra. For example, GL(n, ℂ) is a complex Lie group.


Complex Manifold Maximal Torus Left Translation Local Homeomorphism Flag Manifold 


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

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

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