# Lectures on Selected Topics in Mathematical Physics

- Further applications of Lie theory

- William A Schwalm

- March 2019

### Description

This book is the 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 the solution of certain types of physical problems. It is an informal treatment of these topics, which is intended for physics graduate students or others with a physics background who want a brief and informal introduction to the subjects addressed, in a style and vocabulary that is not completely unfamiliar.

### About Editors

William A Schwalm has been with the Department of Physics and Astrophysics at the University of North Dakota since 1980. His research is in condensed matter theory and application of mathematical methods to physical problems. He is also the author of*Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals*and

*Lectures on Selected Topics in Mathematical Physics: Introduction to Lie Theory with Applications.*

### Table of Contents

Contents

1 Generating Functions

1.1 The basic idea

1.2 Elementary examples of generating functions

2 Groups

2.1 Introduction

2.2 Groups in general and nite groups in particular

2.3 Continuous groups

2.4 Group action and innitesimal generators

2.5 Three examples of generating functions from one-parameter groups

2.6 Quantum oscillator example

2.7 Bessel functions by factoring

2.8 Bessel function generator

2.9 Multiparameter Lie groups

3 Lie algebras

3.1 Algebras

3.2 Associative algebras are essentially matrix algebras

3.3 Lie algebras are commutator sub algebras

3.4 Ideals and classication of complex Lie algebras

3.5 Levi's decomposition

3.6 The Killing form

3.7 Cartan subalgebra

3.8 Root geometry, Weyl group and brief comment on classication

3.9 Representations and Casimir operators

4 Examples and Applications

4.1 The algebra so(5)

4.2 2D Oscillator in a magnetic eld

4.3 Generating functions for spherical harmonics

5 Concluding Remarks

### Bibliographic

Paperback ISBN: 9780750329705

Ebook ISBN: 9781643273495

DOI: 10.1088/2053-2571/aae789

Publisher: Morgan & Claypool Publishers