《李群，李代數(shù)及其表示》是一部學習李群，李代數(shù)及其表示論的優(yōu)秀的研究生教材。與其他一些同類著作相比，《李群，李代數(shù)及其表示》有兩大特點，第一大特點是：作者以一種盡可能少地運用流形知識的方法來研究李群。這種方法十分清晰易懂，使讀者可以快速地掌握知識的核心內容。第二大特點是：《李群，李代數(shù)及其表示》在給出半單李群及李代數(shù)的理論框架之前，通過詳盡地介紹SU（2）和SU（3）的表示理論來引入即將介紹的一般內容，這種方式使得讀者能夠在了解一般理論之前已經(jīng)有了對根系、權，及Weyl群的簡單認識。同時，書中眾多的例子和圖示可以很好地協(xié)助學習并理解一些內容?！独钊?，李代數(shù)及其表示》分為兩部分，第一部分主要介紹了李群與李代數(shù)，以及它們之間的相互關系，同時還介紹了基礎的表示論。第二部分則闡述了半單李群與李代數(shù)理論。This book is intended for a one year graduate course on Lie groups and Lie algebras. The author proceeds beyond the representation theory of compact Lie groups （which is the basis of many texts） and provides a carefully chosen range of material to give the student the bigger picture. For compact Lie groups， the Peter-Weyl theorem， conjugacy of maximal tori （two proofs）， Weyl character formula and more are covered. The book continues with the study of complex analytic groups， then general noncompact Lie groups， including the Coxeter presentation of the Weyl group， the Iwasawa and Bruhat decompositions， Cartan decomposition， symmetric spaces， Cayley transforms， relative root systems， Satake diagrams， extended Dynkin diagrams and a survey of the ways Lie groups may be embedded in one another. The book culminates in a "topics" section giving depth to the students understanding of representation theory， taking the Frobenius-Schur duality between the representation theory of the symmetric group and the unitary groups as a unifying theme， with many applications in diverse areas such as random matrix theory， minors of Toeplitz matrices， symmetric algebra decompositions， Gelfand pairs， Hecke algebras， representations of finite general linear groups and the cohomology of Grassmannians and flag varieties.
作者簡介
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圖書目錄
Preface Chapter 1 Differentiable and Analytic Manifolds 1.1 Differentiable Manifolds 1.2 Analytic Manifolds 1.3 The Frobcnius Theorem 1.4 Appendix Exercises Chapter 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 Lies Theory of Transformation Groups Exercises Chapter 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 Cartans 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 Exercises Chapter 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 …… Bibliogrphy Index
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