Representations of Compact Lie Groups

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Springer Science & Business Media, Jan 1, 1985 - Mathematics - 313 pages
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This book is based on several courses given by the authors since 1966. It introduces the reader to the representation theory of compact Lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. Lie algebras, though mentioned occasionally, are not used in an essential way. The material as well as its presentation are classical; one might say that the foundations were known to Hermann Weyl at least 50 years ago. Prerequisites to the book are standard linear algebra and analysis, including Stokes' theorem for manifolds. The book can be read by German students in their third year, or by first-year graduate students in the United States. Generally speaking the book should be useful for mathematicians with geometric interests and, we hope, for physicists. At the end of each section the reader will find a set of exercises. These vary in character: Some ask the reader to verify statements used in the text, some contain additional information, and some present examples and counter examples. We advise the reader at least to read through the exercises.
  

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Contents

Lie Groups and Lie Algebras
1
2 LeftInvariant Vector Fields and OneParameter Groups
11
3 The Exponential Map
22
4 Homogeneous Spaces and Quotient Groups
30
5 Invariant Integration
40
6 Clifford Algebras and Spinor Groups
54
Elementary Representation Theory
64
1 Representations
65
8 The Complexification of Compact Lie Groups
151
The Maximal Torus of a Compact Lie Group
157
2 Consequences of the Conjugation Theorem
164
3 The Maximal Tori and Weyl Groups of the Classical Groups
169
4 Cartan Subgroups of Nonconnected Compact Groups
176
Root Systems
183
2 Roots and Weyl Chambers
189
3 Root Systems
197

2 Semisimple Modules
72
3 Linear Algebra and Representations
74
4 Characters and Orthogonality Relations
77
5 Representations of SU2 SO3 U2 and O3
84
6 Real and Quaternionic Representations
93
7 The Character Ring and the Representation Ring
102
8 Representations of Abelian Groups
107
9 Representations and Lie Algebras
111
10 The Lie Algebra sl 2 C
115
Representative Functions
123
2 Some Analysis on Compact Groups
129
3 The Theorem of Peter and Weyl
133
4 Applications of the Theorem of Peter and Weyl
136
5 Generalizations of the Theorem of Peter and Weyl
138
6 Induced Representations
143
7 TannakaKrem Duality
146
4 Bases and Weyl Chambers
202
5 Dynkin Diagrams
209
6 The Roots of the Classical Groups
216
7 The Fundamental Group the Center and the Stiefel Diagram
223
8 The Structure of the Compact Groups
232
Irreducible Characters and Weights
239
2 The Dominant Weight and the Structure of the Representation Ring
249
3 The Multiplicities of the Weights of an Irreducible Representation
257
4 Representations of Real or Quaternionic Type
261
5 Representations of the Classical Groups
265
6 Representations of the Spinor Groups
278
7 Representations of the Orthogonal Groups
292
Bibliography
299
Symbol Index
305
Subject Index
307
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