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Lie Groups pp 31-37 | Cite as

Lie Subgroups of \(\mathrm{GL}(n, \mathbb{C})\)

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

Abstract

If U is an open subset of \({\mathbb{R}}^{n}\), we say that a map \(\phi: U\longrightarrow {\mathbb{R}}^{m}\) is smooth if it has continuous partial derivatives of all orders. More generally, if \(X \subset {\mathbb{R}}^{n}\) is not necessarily open, we say that a map \(\phi: X\longrightarrow {\mathbb{R}}^{n}\) is smooth if for each xX there exists an open set U of \({\mathbb{R}}^{n}\) containing \(x\) such that ϕ can be extended to a smooth map on U. A diffeomorphism of \(X \subseteq {\mathbb{R}}^{n}\) with \(Y \subseteq {\mathbb{R}}^{m}\) is a homeomorphism \(F: X\longrightarrow Y\) such that both F and F −1 are smooth. We will assume as known the following useful criterion.

Keywords

Open Subset Topological Group Associative Algebra Jacobi Identity Closed Subgroup 
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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