Lie groups and Lie algebras

mathematics
physics

elective courses

Classical linear groups, abstract Lie groups, compact groups.

Classical Lie's theory: correspondence between Lie groups and Lie algebras. The exponential map.

Abstract approach to Lie algebras. Classification of simple Lie algebras.

Representations of classical Lie groups and Lie algebras by the highest weight. Homogeneous spaces.

1. Examples of classical Lie groups. Quaternions and the symplectic group.

2. Abstract Lie groups. Left-invariant vector fields, the exponential map, the adjoint representation.

3. Tori and their representations. The maximal tori in a compact Lie group.

4. Lie algebras associated to Lie groups. Classical matrix examples.

5. Classical Lie's theory: correspondence between Lie groups and Lie algebras.

6. Abstract approach to Lie algebras. Ideals, quotient Lie algebras and the corresponding group constructions. Relations between properties of Lie groups and Lie algebras.

7. Solvable, nilpotent and semisimple Lie algebras. Killing's form. Cartan's criteria fo solvability and semisimplicity.

8. Properties of Lie algebras associated to compact Lie groups. Invariant bilinear forms. Complex reductive Lie groups (as complexifications of compact Lie groups).

9. Classification of simple Lie algebras by root systems.

10. Representations of compact Lie groups. Characters of representations.

11. Representations of classical Lie groups and Lie algebras. Highest weight representations.

12. Representations of GL (n;C). Young diagrams (information about Pieri formula and Weyl's character formula).

13. Homogeneous spaces of classical groups. Torus action on G/P, fixed points, cell decompostion (using examples of Grassmannian and the flag variety).

1. Adams, J.F. Lectures on Lie groups. 1969

2. Brocker, Theodor; tom Dieck, Tammo. Representations of compact Lie groups. GTM 98, 1985

3. Erdmann K., Wildon M. J. Introduction to Lie Algebras. 2006

4. Fulton, William, Harris, Joe. Representation theory. A rst course. 1991

5. Jacobson, Nathan. Lie algebras. 1962 (1979).

6. Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examplesconstitutes

1986 (2001).

7. Kirillov, Alexander, Jr. An introduction to Lie groups and Lie algebras. Cambridge Studies in Advanced Mathematics, 113. (2008)

Student knows basic notions of Lie group and Lie algebras theory and related representation theory. In particular the student is fluent in the theory presented in the description of the lecture. This constitutes a star of further development and independent research.

The lecture ends with a written and an oral exam. 20 % of the final grade consists of homework and active participation in the exercise sessions.