本书讲述了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。本书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，本书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：但是本书所讲述的是李群和李代数基础理论。

本书内容先进，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。

本书在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，本书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。

Part I. Basic Ideas and Examples

 1. Real and Complex Matrix Groups 3

1.1 Groups of Matrices 3

1.2 Groups of Matrices as Metric Spaces 5

1.3 Compactness. 12

1.4 Matrix Groups 15

1.5 Some Important Examples. 18

1.6 Complex Matrices as Real Matrices. 29

1.7 Continuous Homomorphisms of Matrix Groups 31

1.8 Matrix Groups for Normed Vector Spaces 33

1.9 Continuous Group Actions 37

 2. Exponentials, Differential Equations and One-parameter Subgroups 45

2.1 The Matrix Exponential and Logarithm. 45

2.2 Calculating Exponentials and Jordan Form 51

2.3 Differential Equations in Matrices 55

2.4 One-parameter Subgroups in Matrix Groups 56

2.5 One-parameter Subgroups and Differential Equations 50

 3. Tangent Spaces and Lie Algebras 67

3.1 LieAlgebras 67

8.2 Curves, Tangent Spaces and Lie Algebras 71

3.3 The Lie Algebras of Some Matrix Groups 76                                                    I  I                                                                I            I                   I    I I                                  II

3.4 Some Observations on the Exponential Function of a Matrix Group 84

  3.5 so(3) SU(2) 86

3.6 The Complexification of a Real Lie Algebra 92

 4. Algebras, Quaternions and Quaternionic Symplectic Groups

4.1 Algebras 99

4.2 Real and Complex Normed Algebras 111

4.3 Linear Algebra over a Division Algebra 113

4.4 The Quaternions 116

4.5 Quaternionic Matrix Groups 120                                                                                                                                                                                                                                                 AA

4.6 Automorphism Groups of Algebras 122

 5. Clifford Algebras and Spinor Groups 129

5.1 Real Clifford Algebras 130

5.2 Clifford Groups 139

5.3 Pinor and Spinor Groups 143

5.4 The Centres of Spinor Groups 151

5.5 Finite Subgroups of Spinor Groups 152

 6. Lorentz Groups 157

6.1 Lorentz-Groups 157

6.2 A Principal Axis Theorem for Lorentz Groups 165

6.3 SL2(C) and the Lorentz Group Lor(3, 1) 171

Part II. Matrix Groups as Lie Groups

 7. Lie Groups 181

7.1 Smooth Manifolds 181

7.2 Tangent Spaces and Derivatives 183

7.3 Lie Groups 187

7.4 Some Examples of Lie Groups 180

7.5 Some Useful Formulae in Matrix Groups 103

7.0 Matrix Groups are Lie Groups 1OO

7.7 Not All Lie Groups are Matrix Groups 203

 8. Homogeneous Spaces 211

8.1 Homogeneous Spaces as Manifolds 211

8.2 Homogeneous Spaces as Orbits 215

8.3 Projective Spaces 217

8.4 Grassmannians 222                                                      I              J

8 5 The Gram-Schmidt Process 224

8.6 Reduced Echelon Form 226

8.7 Real Inner Products 227

8.8 Symplectic Forms 220

 9. Connectivity of Matrix Groups 235

9.1 Connectivity of Manifolds 235

9.2 Examples of Path Connected Matrix Groups 238

9.3 The Path Components of a Lie Group 211

9.4 Another Connectivity Result 241

Part 111. Compact Connected Lie Groups and their Classification

 10. Maximal Tori in Compact Connected Lie Groups 251

10.1 Tori 251

10.2 Maximal Tori in Compact Lie Groups. 255

10.3 The Normaliser and Weyl Group of a Maximal Torus 259

10.4 The Centre of a Compact Connected Lie Group 263

 11. Semi simple Factorisation 267

11.1 An Invariant Inner Product 267

11.2 The Centre and its Lie Algebra 270

11.3 Lie Ideals and the Adjoint Action 272

11 4 Semi-simple Decompositions 276

11.5 Structure of the Adjoint Representation 278

 12. Roots Systems, Weyl Groups and Dynkin Diagrams 289

12.1 Inner Products and Duality 280

12.2 Roots systems and their Weyl groups 201

12.3 Some Examples of Root Systems 203

12.4 The Dynkin Diagram of a Root System 207

12.5 Irreducible- --Dynkin---Diagrams 208

12.6 From Root Systems to Lie Algebras 290

Hints and Solutions to Selected Exercises 303

Bibliography 323

Index 325