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
