Principal Structures and Methods of Representation Theory

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American Mathematical Soc. - Mathematics - 430 pages
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The main topic of this book can be described as the theory of algebraic and topological structures admitting natural representations by operators in vector spaces. These structures include topological algebras, Lie algebras, topological groups, and Lie groups. The book is divided into three parts. Part I surveys general facts for beginners, including linear algebra and functional analysis. Part II considers associative algebras, Lie algebras, topological groups, and Lie groups,along with some aspects of ring theory and the theory of algebraic groups. The author provides a detailed account of classical results in related branches of mathematics, such as invariant integration and Lie's theory of connections between Lie groups and Lie algebras. Part III discusses semisimple Liealgebras and Lie groups, Banach algebras, and quantum groups. This is a useful text for a wide range of specialists, including graduate students and researchers working in mathematical physics and specialists interested in modern representation theory. It is suitable for independent study or supplementary reading. Also available from the AMS by this acclaimed author is Compact Lie Groups and Their Representations.
 

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Contents

Basic Notions
3
Algebraic structures
6
Vector spaces 3 Elements of linear algebra
14
Functional calculus 5 Unitary spaces
26
Tensor products
38
Smodules
44
omments to Chapter 1
47
General Theory
49
Local Lie groups
203
Connected Lie groups
209
Representations of Lie groups
214
Examples and exercises
219
Comments to Chapter 5
224
Special Topics
225
Semisimple Lie Algebras
227
Classification
233

Associative Algebras
51
Semisimple modules
58
Group algebras
64
Systems of generators
70
Tensor algebras
75
Formal series
80
Weyl algebras
86
Elements of ring theory
93
Comments to Chapter 2
98
Lie Algebras
99
Solvable Lie algebras
105
Bilinear forms
109
The algebra Ug
115
Semisimple Lie algebras
120
Free Lie algebras
125
Examples of Lie algebras
130
Comments to Chapter 3
137
Topological Groups
139
Topological vector spaces
145
Topological modules
152
Invariant measures
157
Group algebras
164
Compact groups
170
Solvable groups
175
Algebraic groups
181
Comments to Chapter 4
185
Lie Groups
187
Lie groups
192
Formal groups
198
Verma modules
238
Finitedimensional gmodules
244
The algebra Zg
250
The algebra Fºxtg
256
Comments to Chapter 6
262
Semisimple Lie Groups
263
Compact Lie groups
268
Maximal tori
272
Semisimple Lie groups
277
The algebra AG
283
The classical groups
289
Reduction problems
294
Comments to Chapter 7
300
Banach Algebras
301
The commutative case
307
Spectral theory
312
Calgebras
317
Representations of Calgebras
323
Von Neumann algebras
329
The algebra CG
335
Abelian groups
340
Comments to Chapter 8
346
Quantum Groups
347
Appendix A Root Systems
391
Appendix B Banach Spaces
403
The Algebra BH
413
Bibliography
421
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Page 4 - ... Jordanform representations of square matrices, and to compute functions of a matrix, in particular, exponential functions of a matrix (see Section 2-9, Concluding Remarks1). In Section 2-2 we introduce the concepts of field and linear space over a field. The fields we shall encounter in this book are the field of real numbers, the field of complex numbers, and the field of rational functions. In order to have a representation of a vector in a linear space, we introduce, in Section 2-3, the concept...

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