李群結構和李群幾何

《李群結構和李群幾何》是2015年世界圖書出版公司出版的著作，作者是Joachim、Hilgert。

基本介紹

• 中文名：《李群結構和李群幾何》
• 作者：Joachim、Hilgert
• 出版社：世界圖書出版公司
• 出版時間：2015年07月01日
• ISBN：9787510098468 

內容簡介

《李群結構和李群幾何（英文版）》介紹了李群及其在流形上的作用，它受到廣大數學家和學生的喜愛。《李群結構和李群幾何（英文版）》是在作者1991年寫的教材Lie-Gruppen und Lie-Algebren 的基礎上，介紹了李群的基本原理，書中增加了其過去近20年的教學和研究工作編著的，並且著重強調了微分幾何在該領域中的作用。《李群結構和李群幾何（英文版）》內容豐富，書中大量的練習和選用的提示為學生提供了充分的學習指引。

目錄

Introduction

1．1 Teaching Suggestions

1．2 Fundamental Notation

Part Ⅰ Matrix Groups

2 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

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 Semisimple 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 Function 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 Represent

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 Multilinear 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