Lie Groups Beyond an Introduction
From reviews of the first edition: "The important feature of the present book is that it starts from the beginning (with only a very modest knowledge assumed) and covers all important topics... The book is very carefully organized [and] ends with 20 pages of useful historic comments. Such a comprehensive and carefully written treatment of fundamentals of the theory will certainly be a basic reference and text book in the future." -- Newsletter of the EMS "This is a fundamental book and none, beginner or expert, could afford to ignore it. Some results are really difficult to be found in other monographs, while others are for the first time included in a book." -- Mathematica "Each chapter begins with an excellent summary of the content and ends with an exercise section... This is really an outstanding book, well written and beautifully produced. It is both a graduate text and a monograph, so it can be recommended to graduate students as well as to specialists." -- Publicationes Mathematicae 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.
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abstract Cartan abstract root system Ad(g algebra g analytic subgroup automorphism basis Cartan decomposition Cartan involution Cartan matrix Cartan subalgebra Cayley transform Chapter closed linear group commutes complex semisimple Lie complexification conjugate Corollary corresponding define diagonal dimension dimensional direct sum Dynkin diagram eigenvalues element example finite finite-dimensional representation follows formula function given GL(n Haar measure Hence highest weight ideal identity induction invariant subspace irreducible representation isomorphism Killing form Lemma Let G Lie subalgebra linear Lie algebra linear map multiplication nilpotent noncompact nonzero one-one orthogonal parabolic subalgebra polynomial positive roots Problem Proposition prove real Lie algebra reductive Lie group representation of g restricted roots result right side root vectors semisimple Lie algebra semisimple Lie group shows simple roots simply connected smooth SO(n solvable Sp(n subalgebra of g subgroup of G subset symmetric Theorem unitary vector space Vogan diagram Weyl group