Lie groups : an introduction through linear groups / Wulf Rossmann.

Format:

Book

Author/Creator:

Rossmann, Wulf, 1948-

Series:

Oxford graduate texts in mathematics ; 5.

Oxford graduate texts in mathematics ; 5

Language:

English

Subjects (All):

Lie groups.

Physical Description:

x, 265 pages : illustrations ; 25 cm.

Place of Publication:

Oxford ; New York : Oxford University Press, 2002.

Summary:

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 are 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.

Contents:

1 The exponential map 1

1.1 Vector fields and one-parameter groups of linear transformation 1

1.2 Ad, ad, and d exp 12

1.3 The Campbell

Baker

Hausdorff series 22

2 Lie theory 30

2.1 Linear groups: definitions and examples 30

2.2 The Lie algebra of a linear group 44

2.3 Coordinates on a linear group 53

2.4 Connectedness 61

2.5 The Lie correspondence 66

2.6 Homomorphisms and coverings of linear groups 78

2.7 Closed subgroups 87

3 The classical groups 91

3.1 The classical groups: definitions, connectedness 91

3.2 Cartan subgroups 107

3.3 Roots, weights, reflections 115

3.4 Fundamental groups of the classical groups 121

4 Manifolds, homogeneous spaces, Lie groups 132

4.1 Manifolds 132

4.2 Homogeneous spaces 143

4.3 Lie groups 152

5 Integration 165

5.1 Integration on manifolds 165

5.2 Integration on linear groups and their homogeneous spaces 171

5.3 Weyl's integration formula for U(n) 179

6 Representations 189

6.1 Representations: definitions 189

6.2 Schur's lemma, Peter

Weyl theorem 197

6.3 Characters 205

6.4 Weyl's character formula for U(n) 212

6.5 Representations of Lie algebras 223

6.6 The Borel

Weil theorem for GL(n, C) 232

6.7 Representations of the classical groups 237

Appendix Analytic Functions and Inverse Function Theorem 250.

Notes:

Includes bibliographical references (pages 258-260) and index.

ISBN:

0198596839

OCLC:

47927742