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Author's Preface to the Russian Edition This book is written for advanced students, for predoctoral graduate stu dents, and for professional scientists-mathematicians, physicists, and chemists-who desire to study the foundations of the theory of finite dimensional representations of groups. We suppose that the reader is familiar with linear algebra, with elementary mathematical analysis, and with the theory of analytic functions. All else that is needed for reading this book is set down in the book where it is needed or is provided for by references to standard texts. The first two chapters are devoted to the algebraic aspects of the theory of representations and to representations of finite groups. Later chapters take up the principal facts about representations of topological groups, as well as the theory of Lie groups and Lie algebras and their representations. We have arranged our material to help the reader to master first the easier parts of the theory and later the more difficult. In the author's opinion, however, it is algebra that lies at the heart of the whole theory. To keep the size of the book within reasonable bounds, we have limited ourselves to finite-dimensional representations. The author intends to devote another volume to a more general theory, which includes infinite dimensional representations.
Contents
I Algebraic Foundations of Representation Theory.- §1. Fundamental Concepts of Group Theory.- §2. Fundamental Concepts and the Simplest Propositions of Representation Theory.- II Representations of Finite Groups.- §1. Basic Propositions of the Theory of Representations of Finite Groups.- §2. The Group Algebra of a Finite Group.- §3. Representations of the Symmetric Group.- §4. Induced Representations.- §5. Representations of the Group SL (2, Fq).- III Basic Concepts of the Theory of Representations of Topological Groups.- §1. Topological Spaces.- §2. Topological Groups.- §3. Definition of a Finite-Dimensional Representation of a Topological Group; Examples.- §4. General Definition of a Representation of a Topological Group.- IV Representations of Compact Groups.- §1. Compact Topological Groups.- §2. Representations of Compact Groups.- §3. The Group Algebra of a Compact Group.- V Finite-Dimensional Representations of Connected Solvable Groups; the Theorem of Lie.- §1. Connected Topological Groups.- §2. Solvable and Nilpotent Groups.- §3. Lie's Theorem.- VI Finite-Dimensional Representations of the Full Linear Group.- §1. Some Subgroups of the Group G.- §2. Description of the Irreducible Finite-Dimensional Representations of the Group GL (n, C).- §3. Decomposition of a Finite-Dimensional Representation of the Group GL(n, C) into Irreducible Representations.- VII Finite-Dimensional Representations of the Complex Classical Groups.- §1. The Complex Classical Groups.- §2. Finite-Dimensional Continuous Representations of the Complex Classical Groups.- VIII Covering Spaces and Simply Connected Groups.- §1. Covering Spaces.- §2. Simply Connected Spaces and the Principle of Monodromy.- §3. Covering Groups.- §4. Simple Connectedness of Certain Groups.- IX Basic Concepts of Lie Groups and Lie Algebras.- §1. Analytic Manifolds.- §2. Lie Algebras.- §3. Lie Groups.- X Lie Algebras.- §1. Some Definitions.- §2. Representations of Nilpotent and Solvable Lie Algebras.- §3. Radicals ofa Lie Algebra.- §4. The Theory of Replicas.- §5. The Killing Form; Criteria for Solvability and Semisimplicity of a Lie Algebra.- §6. The Universal Enveloping Algebra of a Lie Algebra.- §7. Semisimple Lie Algebras.- §8. Cartan Subalgebras.- §9. The Structure of Semisimple Lie Algebras.- §10. Classification of Simple Lie Algebras.- §11. The Weyl Group of a Semisimple Lie Algebra.- §12. Linear Representations of Semisimple Complex Lie Algebras.- §13. Characters of Finite-Dimensional Irreducible Representations of a Semisimple Lie Algebra.- §14. Real Forms of Semisimple Complex Lie Algebras.- §15. General Theorems on Lie Algebras.- XI Lie Groups.- §1. The Campbell-Hausdorff Formula.- §2. Cartan's Theorem.- §3. Lie's Third Theorem.- §4. Some Properties of Lie Groups in the Large.- §5. Gauss's Decomposition.- §6. Iwasawa's Decomposition.- §7. The Universal Covering Group of a Semisimple Compact Lie Group.- §8. Complex Semisimple Lie Groups and Their Real Forms.- XII Finite-Dimensional Irreducible Representations of Semisimple Lie Groups.- §1. Representations of Complex Semisimple Lie Groups.- §2. Representations of Real Semisimple Lie Groups.- A: Monographs and Textbooks.- B: Journal Articles.
