Lectures on selected topics in mathematical physics : further applications of Lie theory
Lie groups Lie algebras Mathematical physics
IOP Publishing
2019
EISBN 9781643273501
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 finite groups in particular.
2.3. Continuous groups.
2.4. Group action and infinitesimal 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. Multi-parameter Lie groups
3. Lie algebras.
3.1. Algebras.
3.2. Associative algebras are essentially matrix algebras.
3.3. Lie algebras are commutator subalgebras.
3.4. Ideals and classification 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 classification.
3.9. Representations and Casimir operators
4. Examples and applications.
4.1. The algebra so(5).
4.2. Two-dimensional oscillator in a magnetic field.
4.3. Generating functions for spherical harmonics.
5. Concluding remarks.
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.
1.1. The basic idea.
1.2. Elementary examples of generating functions
2. Groups.
2.1. Introduction.
2.2. Groups in general and finite groups in particular.
2.3. Continuous groups.
2.4. Group action and infinitesimal 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. Multi-parameter Lie groups
3. Lie algebras.
3.1. Algebras.
3.2. Associative algebras are essentially matrix algebras.
3.3. Lie algebras are commutator subalgebras.
3.4. Ideals and classification 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 classification.
3.9. Representations and Casimir operators
4. Examples and applications.
4.1. The algebra so(5).
4.2. Two-dimensional oscillator in a magnetic field.
4.3. Generating functions for spherical harmonics.
5. Concluding remarks.
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.
