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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 ��-Basis
6.3 ��-Basis
6.4 ��'-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