Lie algebras and Lie groups

Lecturer:

Libor Šnobl (gar.)

Tutor:

Libor Šnobl (gar.)

Supervisor:

Department of Physics

Synopsis:

Definitions and properties of Lie groups and Lie algebras. Different types of Lie algebras, root systems and classification of complex simple Lie algebras. Introduction into theory of representations.

Requirements:

Basic knowledge of calculus on manifolds, i.e. the contents of 02GMF1 (Geometric Methods in Physics 1) or 02DRG (Differential equations, symmetries and groups).

Syllabus of lectures:

1. Lie group, Lie algebra and their relation.

2. Exponential map.

3. Subgroups and subalgebras, homogeneous spaces.

4. Universal covering.

5. Lie algebras - basic notions.

6. Killing form.

7. Theorems of Lie and Engel.

8. Cartan´s criteria.

9. Cartan´s subalgebra.

10. Root systems.

11. Classification of complex simple Lie algebras.

12. Representations of simple Lie algebras.

Syllabus of tutorials:

1. Groups GL(n), SL(n), O(n), SO(n), U(n), SU(n), Sp(2n), Af(1).

2. Algebras gl(n), sl(n), o(n), so(n), u(n), su(n), sp(2n), af(1).

3. Connectedness and maximal torii of SU(n), SO(n).

4. Exponential map of sl(2) into SL(2).

5. Classification of Lie algebras up to dimension 3.

6. Killing form of Lie algebras up to dimension 3.

7. Root systems of A_l,B_l,C_l,D_l.

8. Tensor product of representations.

9. Representations of SU(3) and their relevance in particle physics.

Study Objective:

knowledge:

Essential notions and results of the theory of Lie groups and algebras.

abilities:

apply methods of Lie groups theory to physics problems

Study materials:

Key references:

[1] D.H. Sattinger, O.L. Weaver: Lie Groups and Algebras, Springer Verlag 1986.

[2] K. Erdmann, M.J. Wildon: Introduction to Lie Algebras, Springer Verlag 2006.

Recommended references:

[3] H. Samelson: Notes on Lie algebras, Springer Verlag 1990.

[4] R. Gilmore: Lie groups, Physics and Geometry, CUP 2008.

[5] R. Penrose: The Road to Reality: A Complete Guide to the Laws of the Universe, Vintage 2007.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans:

• FNSPE Prospectus MGR level (!)