Translations of Mathematical Monographs 1973; 448 pp; softcover Volume: 40 Reprint/Revision History: reprinted with corrections 1978; fifth printing 1996 ISBN10: 0821815903 ISBN13: 9780821815908 List Price: US$131 Member Price: US$104.80 Order Code: MMONO/40
 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 finitedimensional 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 infinitedimensional representations of semisimple complex Lie algebras. Some aspects of the theory of infinitedimensional representations of Lie groups are presented in a brief survey. Readership 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 Gel'fandCetlin 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 finitedimensonal representations
 Infinitesimal theory (characters, weights, Casimir operators)
 Some problems of spectral analysis for finitedimensional representations
 Appendix I. On infinitedimensional 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
