Lie Groups pp 101-108 | 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 \(\mathcal{U} =\{ (U,\phi )\}\), where each chart \(U \subset M\) is an open set and \(\phi: U\longrightarrow {\mathbb{C}}^{m}\) is a homeomorphism of U onto its image that is assumed to be open in \({\mathbb{C}}^{m}\). It is assumed that the transition functions \(\psi \circ {\phi }^{-1}: \phi (U \cap V )\longrightarrow \psi (U \cap V )\) are holomorphic for any two charts (U,ϕ) 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 \(G \times G\longrightarrow G\) and \(G\longrightarrow G\) are holomorphic. The Lie algebra of a complex Lie group is a complex Lie algebra. For example, \(\mathrm{GL}(n, \mathbb{C})\) is a complex Lie group.


Complex Manifold Real Representation Maximal Torus Flag Manifold Compact Torus 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

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

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