李群结构和李群几何
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作者(德)希尔格特(Joachim Hilgert) 著
出版社世界图书出版公司
ISBN9787510098468
出版时间2015-07
装帧平装
开本其他
定价98元
货号1201294956
上书时间2024-12-02
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作者简介
Joachim Hilgert (J.希尔格特)是靠前知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。
目录
Introduction
1.1Teaching Suggestions
1.2l.lndamental Notation
Part Ⅰ Matrix Groups
Concrete Matrix Groups
2.1The General Linear Group
2.2Groups and Geometry
2.3Quaternionic Matrix Groups
3The Matrix Exponential Function
3.1Smooth Functions Defined by Power Series
3.2Elementary Properties of the Exponential Function
3.3The Logarithm Function
3.4The Baker-Campbell-Dynkin-Hausdorff Formula
4Linear Lie Groups
4.1The Lie Algebra of a Linear Lie Group
4.2Calculating Lie Algebras of Linear Lie Groups
4.3Polar Decomposition of Certain Algebraic Lie Groups
Part Ⅱ Lie Algebras
5Elementary Structure Theory of Lie Algebras
5.1Basic Concepts
5.2Nilpotent Lie Algebras
5.3The Jordan Decomposition
5.4Solvable Lie Algebras
5.5Semisimple Lie Algebras
5.6The Theorems of Levi and Malcev
5.7Reductive Lie Algebras
6Root Decomposition
6.1Cartan Subalgebras
6.2The Classification of Simple sl2(K)-Modules
6.3Root Decompositions of Semisimple Lie Algebras
6.4Abstract Root Systems and Their Weyl Groups
7Representation Theory of Lie Algebras
7.1The Universal Enveloping Algebra
7.2Generators and Relations for Semisimple Lie Algebras
7.3Highest Weight Representations
7.4Ados Theorem
7.5Lie Algebra Cohomology
7.6General Extensions of Lie Algebras
Part Ⅲ Manifolds and Lie Groups
8Smooth Manifolds
8.1Smooth Maps in Several Variables
8.2Smooth Manifolds and Smooth Maps
8.3The Tangent Bundle
8.4Vector Fields
8.5Integral Curves and Local Flows
8.6Submanifolds
9 Basic Lie Theory
9.1Lie Groups and Their Lie Algebras
9.2The Exponential Function of a Lie Group
9.3Closed Subgroups of Lie Groups and Their Lie Algebras
9.4Constructing Lie Group Structures on Groups
9.5Covering Theory for Lie Groups
9.6Arcwise Connected Subgroups and Initial Subgroups
10 Smooth Actions of Lie Groups
10.1 Homogeneous Spaces
10.2 Frame Bundles
10.3 Integration on Manifolds
10.4 Invariant Integration
10.5 Integrating Lie Algebras of Vector Fields
Part Ⅳ Structure Theory of Lie Groups
11Normal Subgroups, Nilpotent and Solvable Lie Groups
11.1 Normalizers, Normal Subgroups, and Semidirect Products
11.2 Commutators, Nilpotent and Solvable Lie Groups
11.3 The Automorphism Group of a Lie Group
12Compact Lie Groups
12.1 Lie Groups with Compact Lie Algebra
12.2 Maximal Tori in Compact Lie Groups
12.3 Linearity of Compact Lie Groups
12.4 Topological Properties
13Semlsimple Lie Groups
13.1 Cartan Decompositions
13.2 Compact Real Forms
13.3 The Iwasawa Decomposition
14General Structure Theory
14.1 Maximal Compact Subgroups
14.2 The Center of a Connected Lie Group
14.3 The Manifold Splitting Theorem
14.4 The Exponential Unction of Solvable Groups
14.5 Dense Integral Subgroups
14.6 Appendix: Finitely Generated Abelian Groups
15Complex Lie Groups
15.1 The Universal Complexification
15.2 Linearly Complex Reductive Lie Groups
15.3 Complex Abelian Lie Groups
15.4 The Automorphism Group of a Complex Lie Group
16Linearity of Lie Groups
16.1 Linearly Real Reductive Lie Groups
16.2 The Existence of Faithful Finite-Dimensional Representations
16.3 Linearity of Complex Lie Groups
17Classical Lie Groups
17.1 Compact Classical Groups
17.2 Noncompact Classical Groups
17.3 More Spin Groups
17.4 Conformal Groups
18Nonconnected Lie Groups
18.1 Extensions of Discrete Groups by Lie Groups
18.2 Coverings of Nonconnected Lie Groups
18.3 Appendix: Group Cohomology
Part VAppendices
ABasic Covering Theory
A.1 The Fundamental Group
A.2 Coverings
BSome MultUinear Algebra
B.1 Tensor Products and Tensor Algebra
B.2 Symmetric and Exterior Products
B.3 Clifford Algebras, Pin and Spin Groups
CSome Functional Analysis
C.1 Bounded Operators
C.2 Hilbert Spaces
C.3 Compact Symmetric Operators on Hilbert Spaces
DHints to Exercises
References
Index
内容摘要
该书介绍了李群及其在流形上的作用，它受到广大数学家和学生的喜爱。 该书是在作者1991年写的教材Lie-Gruppen und Lie-Algebren 的基础上，介绍了李群的基本原理，书中增加了其过去近20年的教学和研究工作编著的，并且着重强调了微分几何在该领域中的作用。该书内容丰富， 书中大量的练习和选用的提示为学生提供了充分的学习指引。
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