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李群结构和李群几何

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作者（德）Joachim Hilgert (J.希尔格特)

出版社世界图书出版公司

ISBN9787510098468

出版时间2016-05

装帧平装

开本16开

定价98元

货号23955286

上书时间2024-10-22

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商品描述: 导语摘要
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。

作者简介
Joachim Hilgert (J.希尔格特)是国际知名学者，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
    1.1  Teaching Suggestions
    1.2  l.lndamental Notation
Part Ⅰ Matrix Groups
    Concrete Matrix Groups
    2.1  The General Linear Group
    2.2  Groups and Geometry
    2.3  Quaternionic Matrix Groups
  3  The Matrix Exponential Function
    3.1  Smooth Functions Defined by Power Series
    3.2  Elementary Properties of the Exponential Function
    3.3  The Logarithm Function
    3.4  The Baker-Campbell-Dynkin-Hausdorff Formula
  4  Linear Lie Groups
    4.1  The Lie Algebra of a Linear Lie Group
    4.2  Calculating Lie Algebras of Linear Lie Groups
    4.3  Polar Decomposition of Certain Algebraic Lie Groups
Part Ⅱ Lie Algebras
  5  Elementary Structure Theory of Lie Algebras
    5.1  Basic Concepts
    5.2  Nilpotent Lie Algebras
    5.3  The Jordan Decomposition
    5.4  Solvable Lie Algebras
    5.5  Semisimple Lie Algebras
    5.6  The Theorems of Levi and Malcev
    5.7  Reductive Lie Algebras
  6  Root Decomposition
    6.1  Cartan Subalgebras
    6.2  The Classification of Simple sl2(K)-Modules
    6.3  Root Decompositions of Semisimple Lie Algebras
    6.4  Abstract Root Systems and Their Weyl Groups
  7  Representation Theory of Lie Algebras
    7.1  The Universal Enveloping Algebra
    7.2  Generators and Relations for Semisimple Lie Algebras
    7.3  Highest Weight Representations
    7.4  Ado's Theorem
    7.5  Lie Algebra Cohomology
    7.6  General Extensions of Lie Algebras
Part Ⅲ Manifolds and Lie Groups
  8  Smooth Manifolds
    8.1  Smooth Maps in Several Variables
    8.2  Smooth Manifolds and Smooth Maps
    8.3  The Tangent Bundle
    8.4  Vector Fields
    8.5  Integral Curves and Local Flows
    8.6  Submanifolds
  9   Basic Lie Theory
    9.1  Lie Groups and Their Lie Algebras
    9.2  The Exponential Function of a Lie Group
    9.3  Closed Subgroups of Lie Groups and Their Lie Algebras
    9.4  Constructing Lie Group Structures on Groups
    9.5  Covering Theory for Lie Groups
    9.6  Arcwise 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
  11  Normal 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
  12  Compact 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
  13  Semlsimple Lie Groups
    13.1 Cartan Decompositions
    13.2 Compact Real Forms
    13.3 The Iwasawa Decomposition
  14  General 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
  15  Complex 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
  16  Linearity 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
  17  Classical Lie Groups
    17.1 Compact Classical Groups
    17.2 Noncompact Classical Groups
    17.3 More Spin Groups
    17.4 Conformal Groups
  18  Nonconnected Lie Groups
    18.1 Extensions of Discrete Groups by Lie Groups
    18.2 Coverings of Nonconnected Lie Groups
    18.3 Appendix: Group Cohomology
Part V  Appendices
  A  Basic Covering Theory
    A.1 The Fundamental Group
    A.2 Coverings
  B  Some MultUinear Algebra
    B.1 Tensor Products and Tensor Algebra
    B.2 Symmetric and Exterior Products
    B.3 Clifford Algebras, Pin and Spin Groups
  C  Some Functional Analysis
    C.1 Bounded Operators
    C.2 Hilbert Spaces
    C.3 Compact Symmetric Operators on Hilbert Spaces
  D  Hints 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。

主编推荐
Joachim Hilgert (J.希尔格特)是国际知名学者，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。

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