矩阵群 : 李群理论基础

Part Ⅰ. Basic Ideas and Examples

1. Real and Complex Matrix Groups

1.1 Groups of Matrices

1.2 Groups of Matrices as Metric Spaces

1.3 Compactness

1.4 Matrix Groups

1.5 Some Important Examples

1.6 Complex Matrices as Real Matrices

1.7 Continuous Homomorphisms of Matrix Groups

1.8 Matrix Groups for Normed Vector Spaces

1.0 Continuous Group Actions

2. Exponentials， Differential Equations and One-parameter Subgroups

2.1 The Matrix Exponential and Logarithm

2.2 Calculating Exponentials and Jordan Form

2.3 Differential Equations in Matrices

2.4 One-parameter Subgroups in Matrix Groups

2.5 One-parameter Subgroups and Differential Equations

3. Tangent Spaces and Lie Algebras

3.1 LieAlgebras.

3.2 Curves， Tangent Spaces and Lie Algebras

3.4 Some Observations on the Exponential Function of a Matrix Group

3.5 SO(3) and SU(2)

3.6 The Complexification of a Real Lie Algebra

4. Algebras， Quaternions and Quaternionic Symplectic Groups

4.1 Algebras

4.2 Real and Complex Normed Algebras

4.3 Linear Algebra over a Division Algebra

4.4 The Quaternions

4.5 Quaternionic Matrix Groups

4.6 Automorphism Groups of Algebras

5. Clifford Algebras and Spinor Groups

5.1 Real Clifford Algebras

5.2 Clifford Groups

5.3 Pinor and Spinor Groups

5.4 The Centres of Spinor Groups

5.5 Finite Subgroups of Spinor Groups

6. Lorentz Groups

6.1 Lorentz Groups

6.2 A Principal Axis Theorem for Lorentz Groups

6.3 SL2(C) and the Lorentz Group Lor(3， 1)

Part Ⅱ. Matrix Groups as Lie Groups

7. Lie Groups

7.1 Smooth Manifolds

7.2 Tangent Spaces and Derivatives

7.3 Lie Groups

7.4 Some Examples of Lie Groups

7.5 Some Useful Formulae in Matrix Groups

7.6 Matrix Groups are Lie Groups

7.7 Not All Lie Groups are Matrix Groups

8. Homogeneous Spaces

8.1 Homogeneous Spaces as Manifolds

8.2 Homogeneous Spaces as Orbits

8.3 Projective Spaces

8.4 Grassmannians

8.5 The Gram-Schmidt Process

8.6 Reduced Echelon Form

8.7 Real Inner Products

8.8 Symplectic Forms

9. Connectivity of Matrix Groups

9.1 Connectivity of Manifolds

9.2 Examples of Path Connected Matrix Groups

9.3 The Path Components of a Lie Group

9.4 Another Connectivity Result

Part Ⅲ. Compact Connected Lie Groups and their Classification

10. Maximal Tori in Compact Connected Lie Groups

10.1 Tori

10.2 Maximal Tori in Compact Lie Groups

10.3 The Normaliser and Weyl Group of a Maximal Torus

10.4 The Centre of a Compact Connected Lie Group

11. Semi-simple Factorisation

11.1 An Invariant Inner Product

11.2 The Centre and its Lie Algebra

11.3 Lie Ideals and the Adjoint Action

11.4 Semi-simple Decompositions

11.5 Structure of the Adjoint Representation

12. Roots Systems， Weyl Groups and Dynkin Diagrams

12.1 Inner Products and Duality

12.2 Roots systems and their Weyl groups

12.3 Some Examples of Root Systems

12.4 The Dynkin Diagram of a Root System

12.5 Irreducible Dynkin Diagrams

12.6 From Root Systems to Lie Algebras

Hints and Solutions to Selected Exercises

Bibliography

Index

本书讲述李群和李代数基础理论，内容先进，讲述方法科学，易于掌握和使用。书中有大量例题和习题（附答案或提示），便于阅读。适合用作大学数学系和物理系高年级本科生选修课教材、研究生课程教材或参考书。

《矩阵群：李群理论基础》内容先进，讲述方法科学，有大量例子和习题，并附有习题解答或提示，易于使用。《矩阵群：李群理论基础》在Springer出版社SUMS系列（大学生数学系列）中是内容最深的一册。在我国，《矩阵群：李群理论基础》适合用作大学数学系和物理系高年级本科生选修课教材、研究生课程教材或参考书。
李群和李代数理论是现代数学和物理学的重要工具，也是比较深刻和难学的理论。各种矩阵群和矩阵代数是李群和李代数最典型和最重要的例子。
从矩阵出发讲述这部分数学知识，既能使学生把握内容实质，又能使学生学会计算和使用，所以这是一本不可多得的好教材，应当鼓励中国的老师用这种方法讲述李群和李代数。
就内容而言，《矩阵群：李群理论基础》材料本质上不超出我国大学线性代数、抽象代数和一般拓扑学的教学内容；但是《矩阵群：李群理论基础》所讲述的是李群和李代数基础理论。