**Translations of Mathematical Monographs**

1973;
448 pp;
Softcover

MSC: Primary 22;
Secondary 17

**Print ISBN: 978-0-8218-1590-8
Product Code: MMONO/40**

List Price: $131.00

AMS Member Price: $104.80

MAA Member Price: $117.90

**Electronic ISBN: 978-1-4704-4455-6
Product Code: MMONO/40.E**

List Price: $131.00

AMS Member Price: $104.80

MAA Member Price: $117.90

# Compact Lie Groups and Their Representations

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*D. P. Želobenko*

The contents of this volume are somewhat different from the traditional connotations of the title. First, the author, bearing in mind the needs of the physicist, has tried to make the exposition as elementary as possible. The need for an elementary exposition has influenced the distribution of the material; the book is divided into three largely independent parts, arranged in order of increasing difficulty. Besides compact Lie groups, groups with other topological structure (“similar” to compact groups in some sense) are considered. Prominent among these are reductive complex Lie groups (including semisimple groups), obtained from compact Lie groups by analytic continuation, and also their real forms (reductive real Lie groups). The theory of finite-dimensional representation for these classes of groups is developed, striving whenever possible to emphasize the “compact origin” of these representations, i.e. their analytic relationship to representations of compact Lie groups. Also studied are infinite-dimensional representations of semisimple complex Lie algebras. Some aspects of the theory of infinite-dimensional representations of Lie groups are presented in a brief survey.

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#### Table of Contents

# Table of Contents

## Compact Lie Groups and Their Representations

- Cover Cover11
- Title page iii4
- Contents v6
- Preface 112
- Topological groups. Lie groups 617
- Linear groups 2637
- Fundamental problems of representation theory 4455
- Compact Lie groups. Global theorem 6879
- The infinitesimal method in representation theory 8798
- Analytic continuation 109120
- Irreducible representations of the group 𝑈(𝑛) 126137
- Tensors and Young diagrams 137148
- Casimir operators 156167
- Indicator systems and the Gel′fand-Cetlin basis 176187
- Characters 210221
- Tensor product of two irreducible representations of 𝑈(𝑛) 224235
- Basic types of Lie algebras and Lie groups 238249
- Classification of compact and reductive Lie algebras 259270
- Compact Lie groups in the large 280291
- Description of irreducible finite-dimensonal representations 311322
- Infinitesimal theory (characters, weights, Casimir operators) 348359
- Some problems of spectral analysis for finite-dimensional representations 375386
- Appendix I. On infinite-dimensional representations of semisimple complex Lie groups 398409
- Appendix II. Elements of the general theory of unitary representations of locally compact groups 409420
- Appendix III. Unitary symmetry in the class of elementary particles 422433
- References 433444
- Subject index 444455
- Back Cover Back Cover1461