《矩陣群——李群理論基礎》是2009年清華大學出版社出版的圖書，作者是Baker，Andrew。本書介紹了李群和李代數基礎理論，內容先進，講述方法科學，易於掌握和使用。
基本介紹
- 書名：矩陣群——李群理論基礎
- 又名：Matrix Groups: An Introduction to Lie Group Theory
- 作者：Baker, Andrew著
- ISBN：9787302214847
- 出版社：清華大學出版社
- 出版時間：2009-11-1
圖書信息,圖書簡介,目錄,
圖書信息
書名：Matrix Groups: An Introduction to Lie Group Theory(矩陣群——李群理論基礎)
ISBN：9787302214847
作者：Baker, Andrew著
定價：45元
出版日期：2009-11-1
出版社：清華大學出版社
圖書簡介
書中有大量例題和習題（附答案或提示），便於閱讀。適合用作大學數學系和物理系高年級本科生選修課教材、研究生課程教材或參考書
目錄
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
