Lie Groups: An Introduction Through Linear Groups

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Oxford University Press, 2006 - Mathematics - 265 pages
This book is an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups and Lie algebras is developed in its local and global aspects. The classical groups are analyzed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, root, weights and reflections. The fundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements of differential and integral calculus on manifolds are presented, with special emphasis on integration on groups and homogeneous spaces. Representation theory starts from first principles, such as Schur's lemma and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.
 

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Contents

Contents
6
Lie theory
30
The classical groups
91
Manifolds homogeneous spaces Lie groups
132
Integration
165
Representations
189
Appendix Analytic Functions and Inverse
250
References
258
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About the author (2006)

Wulf Rossmann is in the Department of Mathematics and Statistics, University of Ottawa.

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