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Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups.
Topics include a description of all simply connected Lie groups in terms of semisimple Lie groups and semidirect products, the Cartan theory of complex semisimple Lie algebras, the Cartan-Weyl theory of the structure and representations of compact Lie groups and representations of complex semisimple Lie algebras, the classification of real semisimple Lie algebras, the structure theory of noncompact reductive Lie groups as it is now used in research, and integration on reductive groups. Many problems, tables, and bibliographical notes complete this comprehensive work, making the text suitable either for self-study or for courses in the second year of graduate study and beyond.
Content Level »Research
Keywords »Algebra/Rings - Group representation - Group theory - Groups & Generalizations - Lie Groups - Math Physics - Mathematics - Representation/Lie Group - Vector space - algebra
Preface * Prerequisites by chapter * Standard Notation * I. Lie Algebras and Lie Groups * II. Complex Semisimple Lie Algebras * III. Universal Enveloping Algebra * IV. Compact Lie Groups * V. Finite-Dimensional Representations * VI. Structure Theory of Semisimple Groups * VII. Advanced Structure Theory * VIII. Integration * IX. Induced Representations and Branching Theorems * X. Prehomogeneous Vector Spaces * Appendices * Hints for Solutions of Problems * Notes * References * Index of Notation * Index