李群,李代數(shù)及其表示

內(nèi)容概要

This book provides an introduction to Lie groups, Lie algebras, and representation theory, aimed at graduate students in mathematics and physics.Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a　useful addition to the literature. First, it treats Lie groups (not just Lie alge bras) in a way that minimizes the amount of manifold theory needed. Thus,I neither assume a prior course on differentiable manifolds nor provide a con-densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semisimple groups and Lie algebras by　treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory.    The standard books on Lie theory begin immediately with the general case:a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time. Furthermore, with this approach, one must either assume the reader is familiar with the theory of differentiable manifolds (which rules out a substantial part of one's audience) or one must spend considerable time at the beginning of the book explaining this theory (in which case, it takes a long time to get to Lie theory proper).

書籍目錄

PrefaceChapter 1  Differentiable and Analytic Manifolds  1.1  Differentiable Manifolds  1.2  Analytic Manifolds  1.3  The Frobcnius Theorem  1.4  Appendix  ExercisesChapter 2  Lie Groups and Lie Algebras  2.1  Definition and Examples of Lie Groups  2.2  Lie Algebras  2.3  The Lie Algebra of a Lie Group  2.4  The Enveloping Algebra of a Lie Group  2.5  Subgroups and Subalgebras  2.6  Locally isomorphic Groups  2.7  Homomorphisms  2.8  The Fundamental Theorem of Lie  2.9  Closed Lie Subgroups and Homogeneous Spaces. Orbits and Spaces of Orbits  2.10 The Exponential Map  2.11  The Uniqueness of the Real Analytic Structure of a Real Lie Group  2.12  Taylor Series Expansions on a Lie Group  2.13  The Adjoint Representations of!~ and G  2.14  The Differential of the Exponential Map  2.15  The Baker-CampbelI-Hausdorff Formula  2.16  Lie's Theory of Transformation Groups  ExercisesChapter 3  Structure Theory  3.1  Review of Linear Algebra　3.2  The Universal Enveloping Algebra of a Lie Algebra  3.3  The Universal Enveloping Algebra as a Filtered Algebra  3.4  The Enveloping Algebra of a Lie Group  3.5  Nilpotent Lie Algebras  3.6  Nilpotent Analytic Groups  3.7  Solvable Lie Algebras  3.8  The Radical and the Nil Radical  3.9  Cartan's Criteria for Solvability and Semisimplicity  3.10  Semisimple Lie Algebras  3.11  The Casimir Element  3.12  Some Cohomology  3.13  The Theorem of Weyl  3.14  The Levi Decomposition  3.15  The Analytic Group of a Lie Algebra  3.16  Reductive Lie Algebras  3.17  The Theorem of Ado  3.18  Some Global Results  ExercisesChapter 4  Complex Semisimple Lie Algebras And Lie Groups: Structure and Representation  4.1  Cartan Subalgebras     4.2  The Representations of t(2, C)  4.3  Structure Theory     4.4  The Classical Lie Algebras  4.5  Determination of the Simple Lie Algebras over C     4.6  Representations with a Highest Weight    4.7  Representations of Semisimple Lie Algebras   4.8  Construction of a Semisimple Lie Algebra from its Cartan Matrix   　……BibliogrphyIndex

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