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李群，李代数及其表示
作者：（美国）（V.S.Varadarajan）范阮达若詹
出版社：世界图书出版公司
出版时间：2008-01-01
ISBN：9787506292245
定价：¥69.00
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内容简介
《李群，李代数及其表示》是一部学习李群，李代数及其表示论的优秀的研究生教材。与其他一些同类著作相比，《李群，李代数及其表示》有两大特点，第一大特点是：作者以一种尽可能少地运用流形知识的方法来研究李群。这种方法十分清晰易懂，使读者可以快速地掌握知识的核心内容。第二大特点是：《李群，李代数及其表示》在给出半单李群及李代数的理论框架之前，通过详尽地介绍SU（2）和SU（3）的表示理论来引入即将介绍的一般内容，这种方式使得读者能够在了解一般理论之前已经有了对根系、权，及Weyl群的简单认识。同时，书中众多的例子和图示可以很好地协助学习并理解一些内容。《李群，李代数及其表示》分为两部分，第一部分主要介绍了李群与李代数，以及它们之间的相互关系，同时还介绍了基础的表示论。第二部分则阐述了半单李群与李代数理论。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.
作者简介
暂缺《李群，李代数及其表示》作者简介
目录
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
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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