Solutions to some exercises, and short review notes (please report mistakes/typos)

Mock exam
Solutions to part B of mock exam


Teaching arrangements:
Two hours of lectures per week: Mondays 12:00 - 14:00, S3.30 [330], 18 Jan 2010 - 29 March (incl.) 2010,
plus one revision lecture, 26 April 2010.

Prerequisites:
Basic knowledge of vector spaces, matrices, groups, real analysis.

Assessment:
One two-hour written examination at the end of the academic year.

Assignments:
Exercises taken from the main notes and books. Solutions will be provided (see above).

Aims and objectives:
This course gives an introduction to the theory of Lie groups, Lie
algebras and their representations. Lie groups are essentially groups
with continuous parameters, in such a way that the elements form a manifold.
They arise in many parts of mathematics and physics, often in the form of
matrices satisfying certain conditions (e.g. that the matrices should be
invertible, or unitary, or orthogonal). One of the beauties of the
subject is the way that methods from many different areas of
mathematics (algebra, geometry, analysis) are all brought in together.
The course should enable you to go on to further topics in
group theory, differential geometry, quantum field theory,
string theory and other areas.

Syllabus:
Definitions of the basic structures: Lie algebras and Lie groups.
Examples of Lie groups and Lie algebras. Matrix Lie groups, their Lie
algebras, the exponential map, Baker-Campbell-Hausdorff formula.
Abstract Lie algebras, examples: sl(2), sl(3), Poincare algebra.
Representations of Lie algebras, sub-representations, Schur's Lemma,
tensor products. Cartan-Weyl basis, classification of simple Lie
algebras (without  proof).

Books:
There is no book that covers all the material exactly as taught, but the course will be mainly inspired by the following:

1. BC Hall: An Elementary Introduction to Groups and Representations. arxiv:math-ph/0005032v1 (for Lie groups and examples)
2. JE Humphreys: Introduction to Lie Algebras and Representation Theory. Springer 1972 (for abstract Lie algebra)

and some additional material will be taken from last year's notes:

3. I Runkel: Lie Groups and Lie Algebras, KCL notes.

Hall's notes above contain a short descriptive bibliography on the subject. There is also an extended version of these notes:

- BC Hall: Lie Groups, Lie Algebras and Representations: An Elementary Introduction. Springer 2003

To this I would add the book:

- R Gilmore: Lie groups, Lie algebras and some of their applications. Krieger 1994

Feb 17, 2010