Group Theory Lecture Notes

Notational Conventions Introduction, Definitions and Examples Basic Definitions and Terminology Subgroups and Cosets Direct Product of Groups Cyclic, Dihedral and Permutation Groups Cyclic Group Dihedral Group Permutation Group Orbit Stabiliser Theorem Further Definitions Automorphisms and Semi-Direct Product Wreath Products and Central Products Conjugacy Classes Normaliser, Centraliser and Commutator Double Coset Goursat's Lemma Quaternion Groups Tetrahedral, Octahedral and Icosahedral Groups Matrix Groups Orthogonal Unitary Symplectic Heisenberg Group Compact and Non Compact Representations Schur's Lemmas Induced Representations Unitary Representations Infinite Dimensional Unitary Representations Real and Pseudo-Real Representations Orthogonality Relations Characters Further Constraints on Dimensions of Irreducible Representations Tensor Products Symmetric and Antisymmetric Products Character Tables Molien Series Anticommuting Molien Series Examples of Molien Series Molien Series and Wreath Products Symmetries in Quantum Mechanics, Projective and Anti-Unitary Representations Projective Representations Anti-Unitary Representations Rotations and Angular Momentum, SO(3) and SU(2) Three Dimensional Rotations Isomorphism of SO(3) and SU(2)/Z2 Non Compact Isomorphisms Infinitesimal Rotations and Generators Representations of Angular Momentum Commutation Relations The |j m "5365365 basis Action of Time Reversal Representation Matrices Integration over SO(3) and orthogonality relations Characters for SU(2) Tensor Products and Angular Momentum Addition Examples of the calculation of Clebsch-Gordan coefficients Construction of Singlet States Construction of Highest Weight States Special Cases of Clebsch-Gordan Coefficients 3j Symbol 6j Symbol Crossing Relations Tensor Products and Characters SO(3) Tensors Spherical Harmonics Molien Series for SU(2) Irreducible Tensor Operators Spinors Spinor Representation for Angular Momentum Calculation of Action of Derivatives Isospin G-parity Relativistic Symmetries, Lorentz and Poincaré Groups Lorentz Group Proof of Linearity Structure of Lorentz Group Infinitesimal Lorentz Transformations and Commutation Relations Lorentz Group and Spinors Isomorphism SO(3,1) Sl(2,C)/Z2 Spinors, Dotted and Undotted Indices Tensorial Representations Poincaré Group Irreducible Representations of the Poincaré Group Massive Representations Helicity States Massless Representations Two Particle States and Angular Momentum Decomposition Spinorial Treatment Casimir Operators Quantum Fields Lie Groups and Lie Algebras Vector Fields, Differential Forms and Lie Brackets Lie Algebras Lie Algebra Definitions Matrix Lie Algebras and Matrix Lie Groups SU(2) Example Upper Triangular Matrices Representations and Lie Algebras Relation of Lie Algebras to Lie Groups One-Parameter Subgroups Baker Cambell Hausdorff Formula Simply Connected Lie Groups and Covering Groups Covering Group Projective Representations Lie Algebra and Projective Representations Galilean Group Integration over a Lie Group, Compactness SU(2) Example Non Compact Sl(2,R) Example Adjoint Representation and its Corollaries Killing Form Conditions for Non Degenerate Killing Form Decomposition of Semi-simple Lie Algebras Casimir Operators and Central Extensions Lie Algebras for Matrix Groups Unitary Groups Symplectic Groups Orthogonal and Spin Groups Spin Groups and Gamma Matrices Products and Traces of Gamma Matrices Construction of Representations of the Clifford Algebra Conjugation Matrix for Gamma Matrices Special Cases Fierz Identities SU(3) and its Representations Recap of su(2) A su(3) Lie algebra basis and its automorphisms Highest Weight Representations for su(3) Analysis of the Weight Diagram SU(3) Characters Casimir operator Particular SU(3) Representations SU(3) Tensor Representations su(3) Lie algebra again SU(3) and Physics SU(3)F Symmetry Breaking SU(3) and Colour Tensor Products for SU(3) Systematic Discussion of Tensor Products Gauge Groups and Gauge Theories Abelian Gauge Theories Non Abelian Gauge Theories Chern-Simons Theory Gauge Invariants and Wilson Loops Wilson Loops Integrations over Spaces Modulo Group Transformations Integrals over Spheres Integrals over Symmetric and Hermitian Matrices Large n Limits Integrals over Compact Matrix Groups Integration over a Gauge Field and Gauge Fixing