标题：矩阵群——李群理论基础
图书信息
书名：Matrix Groups: An Introduction to Lie Group Theory(矩阵群——李群理论基础)　　
ISBN：9787302214847　　
作者：Baker, Andrew著　　
定价：45元　　
出版日期：2009-11-1　　
出版社：清华大学出版社
图书简介
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
目录
Contents
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.O  Continuous Group Actions................................. 37
2.  Exponentials, Differential Equations and One-parameter Sub-
groups...................................................... 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
8.  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
OWO
xiv                                         Contents
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 ~O
4.1  Algebras ................................................ OO
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
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4.{j  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 ..........................................-1-~57
6.2  A Principal Axis Theorem for Lorentz Groups ............... 165
0.3 SL2(C) and the Lorentz Group Lor(3, 1)..................... ltl
II                   I    I I                                 I I                                  I
Part II. Matrix Groups as Lie Groups
I I                                     L      I                                              II     I                                                mmllm  !   i
7.  Lie Groups ................................................. 181
-7.-1  Smooth Manifolds ........................................-1-8-1
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
Contents                                            xv
II    II I I I                           J
8 5  The Gram-Schmidt Process                      224
~ ~        ~        ~        ~        S        S        ~        ~        S        ~        ~        ~        S        ~        ~        ~        ~        ~        S        ~        ~        ~        ~        ~        ~        ~        ~        ~        ~        ~        ~
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 ................................. 2315
0.2  Examples of Path Connected Matrix Groups................. 238
0.3  The Path Components of a Lie Group....................... 2,11
0.4  Another Connectivity Result............................... 24,1
Part 111. Compact Connected Lie Groups and their Classification
I                I I I
10. Maximal Tori in Compact Connected Lie Groups........... 251
10.1 Tori .................................................... 2151
10.2 Maximal Tori in Compact Lie Groups....................... 2155
10.3 The Normaliser and Weyl Group of a Maximal Torus ......... 2150
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.:1 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
-1-2.-,5 Irreducible- --Dynkin---Diagrams............................... 208
12.6 From Root Systems to Lie Algebras......................... 290
Hints and Solutions to Selected Exercises....................... 303
Bibliography                                  323
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Index........................................................... 325
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