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Lie Groups pp 101-108 | Cite as

Tori

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

Abstract

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.

Keywords

Complex Manifold Real Representation Maximal Torus Flag Manifold Compact Torus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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