Group Theory

Abstract

The article conbtains sections titled: Introduction Precursors to Group Theory Classical Geometry Dimensional Analysis Scaling Dynamical Similarity Groups: Definitions Group Axioms Isomorphisms and Homomorphisms Examples of Discrete Groups Finite Groups Infinite Discrete Groups Examples of Matrix Groups Translation Groups Heisenberg Group Rotation Group Lorentz Group Lie Groups Lie Algebras Structure Constants Constructing Lie Algebras by Linearization Constructing Lie Groups by Exponentiation Cartan Metric Operator Realizations of Lie Algebras Disentangling Results Riemannian Symmetric Spaces Applications in Classical Physics Principle of Relativity Making Mechanics and Electrodynamics Compatible Gravitation Reflections Linear Representations Maps to Matrices Group Element–Matrix Element Duality Classes and Characters Fourier Analysis on Groups Irreps of Crystal Field Theory Symmetry Groups Dynamical Groups Conformal Symmetry Atomic Shell Structure Nuclear Shell Structure Dynamical Models Gauge Theory Group Theory and Special Functions Summary of Some Properties Relation with Lie Groups Spherical Harmonics and Differential and Recursion Relations Differential Equation Addition Theorems Generating Functions Summary Glossary

An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics

“…with x = (x 1 , x 2 , x 3 ) ∈ R 3 [3]. It is easy to verify that φ and ψ are differentiable maps, and so H 3 is indeed a Lie group.…”

“…Lie groups are continuous groups with differentiable composition and inversion maps. Physicists are most familiar with Lie groups having matrix representations, particularly in particle physics, e.g., SU (2) or SU (3). It was discovered that many of the special functions are matrix elements of matrix representations of different Lie groups.…”