Schwalm W.A. Lectures on Selected Topics in Mathematical Physics: Further Applications of Lie Theory

Bristol: IOP Publishing, 2019. — 114 p.This book is a sequel to Lectures on Selected Topics in Mathematical Physics: Introduction to Lie theory with applications. This volume is devoted mostly to Lie groups. Lie algebras and generating functions, both for standard special functions and for solution of certain types of physical problems. It is an informal treatment of these topics intended for physics graduate students or others with a physics background wanting a brief and informal introduction to the subjects addressed in a style and vocabulary not completely unfamiliar.Acknowledgements
Author Biography
William A SchwalmGenerating functions
The basic idea
Elementary examples of generating functionsGroupsGroups in general and finite groups in particular
Continuous groups
Group action and infinitesimal generators
Three examples of generating functions from one-parameter groups
Quantum oscillator example
Bessel functions by factoring
Bessel function generator
Multi-parameter Lie groupsLie algebras
Algebras
Associative algs are essentially matrix algebras
Lie algebras are commutator subalgebras
Ideals and classification of complex Lie algebras
Levi’s decomposition
The Killing form
Cartan subalgebra
Root geometry, Weyl group and brief comment on classification
Representations and Casimir operatorsExamples and applications
The algebra so(5)
Two-dimensional oscillator in a magnetic field
Generating functions for spherical harmonicsConcluding remarks