群论

1 Preface： the pursuit of symmetries

2 Finite groups： an introduction

2.1 Group axioms

2.2 Finite groups of low order

2.3 Permutations

2.4 Basic concepts

2.4.1 Conjugation

2.4.2 Simple groups

2.4.3 Sylow's criteria

2.4.4 Semi-direct product

2.4.5 Young Tableaux

3 Finite groups： representations

3.1 Introduction

3.2 Schur's lemmas

3.3 The ，，44 character table

3.4 Kronecker products

3.5 Real and complex representations

3.6 Embeddings

3.7 Zn character table

3.8 Dn character table

3.9 Q2， character table

3.10 Some semi-direct products

3.11 Induced representations

3.12 Invariants

3.13 Coverings

4 Hilbert spaces

4.1 Finite Hilbert spaces

4.2 Fermi oscillators

4.3 Infinite Hilbert spaces

5 SU（2）

5.1 Introduction

5.2 Some representations

5.3 From Lie algebras to Lie groups

5.4 SU（2） → SU（1， 1）

5.5 Selected SU（2） applications

5.5.1 The isotropic harmonic oscillator

5.5.2 The Bohr atom

5.5.3 Isotopic spin

6 SU（3）

6.1 SU（3） algebra

6.2 α-Basis

6.3 β-Basis

6.4 α'-Basis

6.5 The triplet representation

6.6 The Chevalley basis

6.7 SU（3） in physics

6.7.1 The isotropic harmonic oscillator redux

6.7.2 The Elliott model

6.7.3 The Sakata model

6.7.4 The Eightfold Way

7 Classification of compact simple Lie algebras

7.1 Classification

7.2 Simple roots

7.3 Rank-two algebras

7.4 Dynkin diagrams

7.5 Orthonormal bases

8 Lie algebras： representation theory

8.1 Representation basics

8.2 A3 fundamentals

8.3 The Weyl group

8.4 Orthogonal Lie algebras

8.5 Spinor representations

8.5.1 SO（2n） spinors

8.5.2 SO（2n + 1） spinors

8.5.3 Clifford algebra construction

8.6 Casimir invariants and Dynkin indices

8.7 Embeddings

8.8 Oscillator representations

8.9 Verma modules

8.9.1 Weyl dimension formula

8.9.2 Verma basis

9 Finite groups： the road to simplicity

9.1 Matrices over Galois fields

9.1.1 PSL2（7）

9.1.2 A doubly transitive group

9.2 Chevalley groups

9.3 A fleeting glimpse at the sporadic groups

10 Beyond Lie algebras

10.1 Serre presentation

10.2 Affine Kac-Moody algebras

10.3 Super algebras

11 The groups of the Standard Model

11.1 Space-time symmetries

11.1.1 The Lorentz and Poincar6 groups

11.1.2 The conformal group

11.2 Beyond space-time symmetries

11.2.1 Color and the quark model

11.3 Invariant Lagrangians

11.4 Non-Abelian gauge theories

11.5 The Standard Model

11.6 Grand Unification

11.7 Possible family symmetries

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

11.7.2 Finite SU（3） subgroups

12 Exceptional structures

12.1 Hurwitz algebras

12.2 Matrices over Hurwitz algebras

12.3 The Magic Square

Appendix 1 Properties of some finite groups

Appendix 2 Properties of selected Lie algebras

References

Index

《群论》旨在为物理学家介绍群理论的许多有趣的数学方面，同时将数学家带入物理应用。针对高年级本科生和研究生，书中给出了有限群和连续群的最全面的特点，并且强调在基础物理中的应用；展开讨论了有限群，重点强调了不可约表示和不变性；详细论述了李群，也用较多的笔墨讲述了Kac－Moody代数，包括Dynkin图。