群论 英文PDF电子书下载

1 Preface:the pursuit of symmetries 1

2 Finite groups:an introduction 4

2.1 Group axioms 5

2.2 Finite groups of low order 6

2.3 Permutations 19

2.4 Basic concepts 22

2.4.1 Conjugation 22

2.4.2 Simple groups 25

2.4.3 Sylow's criteria 27

2.4.4 Semi-direct product 28

2.4.5 Young Tableaux 31

3 Finite groups:representations 33

3.1 Introduction 33

3.2 Schur's lemmas 35

3.3 The A4 character table 41

3.4 Kronecker products 44

3.5 Real and complex representations 46

3.6 Embeddings 48

3.7 Zn character table 52

3.8 Dn character table 53

3.9 Q2n character table 56

3.10 Some semi-direct products 58

3.11 Induced representations 61

3.12 Invariants 64

3.13 Coverings 67

4 Hilbert spaces 69

4.1 Finite Hilbert spaces 69

4.2 Fermi oscillators 70

4.3 Infinite Hilbert spaces 72

5 SU(2) 78

5.1 Introduction 78

5.2 Some representations 82

5.3 From Lie algebras to Lie groups 86

5.4 SU(2)→SU(1,1) 89

5.5 Selected SU(2)applications 93

5.5.1 The isotropic harmonic oscillator 93

5.5.2 The Bohr atom 95

5.5.3 Isotopic spin 99

6 SU(3) 102

6.1 SU(3)algebra 102

6.2 α-Basis 106

6.3 ω-Basis 107

6.4 α′-Basis 108

6.5 The triplet representation 110

6.6 The Chevalley basis 112

6.7 SU(3)in physics 114

6.7.1 The isotropic harmonic oscillator redux 114

6.7.2 The Elliott model 115

6.7.3 The Sakata model 117

6.7.4 The Eightfold Way 118

7 Classification of compact simple Lie algebras 123

7.1 Classification 124

7.2 Simple roots 129

7.3 Rank-two algebras 131

7.4 Dynkin diagrams 134

7.5 Orthonormal bases 140

8 Lie algebras:representation theory 143

8.1 Representation basics 143

8.2 A3 fundamentals 144

8.3 The Weyl group 149

8.4 Orthogonal Lie algebras 151

8.5 Spinor representations 153

8.5.1 SO(2n)spinors 154

8.5.2 SO(2n+1)spinors 156

8.5.3 Clifford algebra construction 159

8.6 Casimir invariants and Dynkin indices 164

8.7 Embeddings 168

8.8 Oscillator representations 178

8.9 Verma modules 180

8.9.1 Weyl dimension formula 187

8.9.2 Verma basis 188

9 Finite groups:the road to simplicity 190

9.1 Matrices over Galois fields 192

9.1.1 PSL2(7) 197

9.1.2 A doubly transitive group 198

9.2 Chevalley groups 201

9.3 A fleeting glimpse at the sporadic groups 205

10 Beyond Lie algebras 208

10.1 Serre presentation 208

10.2 Affine Kac-Moody algebras 210

10.3 Super algebras 216

11 The groups of the Standard Model 221

11.1 Space-time symmetries 222

11.1.1 The Lorentz and Poincaré groups 223

11.1.2 The conformal group 231

11.2 Beyond space-time symmetries 235

11.2.1 Color and the quark model 239

11.3 Invariant Lagrangians 240

11.4 Non-Abelian gauge theories 243

11.5 The Standard Model 244

11.6 Grand Unification 246

11.7 Possible family symmetries 249

11.7.1 Finite SU(2)and SO(3)subgroups 249

11.7.2 Finite SU(3)subgroups 252

12 Exceptional structures 254

12.1 Hurwitz algebras 254

12.2 Matrices over Hurwitz algebras 257

12.3 The Magic Square 259

Appendix 1 Properties of some finite groups 265

Appendix 2 Properties of selected Lie algebras 277

References 307

Index 308