F11LG - Lie Groups and Lie Algebra

Course leader(s):

Robert Andrew Weston (Edinburgh, January)

Aims

The purpose of this course is to introduce the concepts of Lie groups and Lie algebras and to familiarise the students with the basics of Lie theory. Furthermore, students will encounter applications of Lie groups and Lie algebras in a number of different contexts. In addition to the material covered in F10LG, this course gives an introduction to the basics of the representation theory of Lie groups and Lie algebras.

Syllabus

1. Introductory Material (1.1 1. Motivation, 1.2 2. A loose characterization of Lie Groups and Lie Algebras, 1.3 3. Examples)

2. Matrix Group Basics (2.1 1. Groups: review of basic definitions, 2.2 2. Matrix groups and classical groups, 2.3 3. Products of groups)

3. Matrix Lie groups (3.1 1. Key concepts, 3.2 2. Matrix Lie Groups, 3.3 3. Connected Matrix Lie Groups, 3.4 4. Compact Matrix Lie Groups)

4. Applications of Matrix Lie Groups (4.1 1. The Lorentz Group, 4.2 2. Quarternions)

5. The Matrix Exponential (5.1 1. Motivation, 5.2 2. The matrix exponential, 5.3 3. The matrix logarithm, 5.4 4. The Baker-Campbell-Haussdorff formula)

6. Lie Algebras (6.1 1. Motivation, 6.2 2. Definition of a matrix Lie algebras, 6.3 3. Examples of matrix Lie algebras, 6.4 3. The abstract definition of a Lie algebra)

7. Matrix Lie groups and Lie algebras (7.1 1. The connection between matrix Lie groups and Lie algebras, 7.2 2. Examples, 7.3 3. Lie group homormorphisms and Lie algebra homomorphisms, 7.4 4. General Lie theory)

8. Representations of Lie groups and Lie algebras (8.1 1. Representation theory basics, 8.2 2. Examples of representations, 8.3 3. The connection between representations of Lie groups and Lie algebras, 8.4 4. Representations of sl2,C, 8.5 5. Representations of sl3,C)

9. Application: Solving Ordinary Differential Equations (9.1 1. Symmetry groups of ODEs, 9.2 2. Lie algebra techniques for solving ODEs)

10. Further representation theory and classification of Lie algebras (10.1 1. Tensor products of representations, 10.2 2. Schur's Lemma, 10.3 3. The Chevalley-Serre basis, 10.4 4. Classification of simple Lie algebras)

Learning outcomes

By the end of the course, students should be able to do the following:

appy the key concepts to the applications of Special Relativity and the Mobius Group
comprehend and use the details of the relation between a Lie group and its Lie algebra
apply basic concepts from topology in the context of matrix Lie groups
apply Lie groups to solve ODEs
use the key concepts of representation theory to construct and manipulate the representations of key examples of Lie algebras
use the fundamental concepts of matrix Lie groups and matrix Lie algebras in order to analyse examples

Further details

Curriculum explorer: Click here

SCQF Level: 11

Credits: 15