Compact Lie Groups and Their Representations

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.

Table of Contents:
Part I. Introduction: Topological groups. Lie groups Linear groups Fundamental problems of representation theory; Part II. Elementary theory: Compact Lie groups. Global theorem The infinitesimal method in representation theory Analytic continuation Irreducible representations of the group $\mathrm {U}(n)$ Tensors and Young diagrams Casimir operators Indicator systems and the Gelfand-Cetlin basis Characters Tensor product of two irreducible representations of $\mathrm {U}(n)$; Part III. General theory: 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; Appexdix III. Unitary symmetry in the class of elementary particles References Subject index.