Theory of Group Representations

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 of a 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.