Compact Lie Groups and Their Representations

✍ Scribed by D. P. Zelobenko

• Publisher: American Mathematical Society
• Year: 1973
• Tongue: English
• Leaves: 461
• Series: Translations of Mathematical Monographs
• Category: Library

✦ Synopsis

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 thecompact 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.

✦ Table of Contents

Cover
Title page
Contents
Preface
Topological groups. Lie groups
Linear groups
Fundamental problems of representation theory
Compact Lie groups. Global theorem
The infinitesimal method in representation theory
Analytic continuation
Irreducible representations of the group 𝑈(𝑛)
Tensors and Young diagrams
Casimir operators
Indicator systems and the Gel′fand-Cetlin basis
Characters
Tensor product of two irreducible representations of 𝑈(𝑛)
Basic types of Lie algebras and Lie groups
Classification of compact and reductive Lie algebras
Compact Lie groups in the large
Description of irreducible finite-dimensonal representations
Infinitesimal theory (characters, weights, Casimir operators)
Some problems of spectral analysis for finite-dimensional representations
Appendix I. On infinite-dimensional representations of semisimple complex Lie groups
Appendix II. Elements of the general theory of unitary representations of locally compact groups
Appendix III. Unitary symmetry in the class of elementary particles
References
Subject index
Back Cover