Representation Theory

4 hours of lectures per week.

Lecturer:

Henning Haahr Andersen

Content:

This will be an introduction to a large field in modern Mathematics. We shall focus on Lie Algebras. These objects are named after the Norwegian mathematician Sophus Lie (1842-1899). They play a rôle in several areas of mathematics. Our treatment will be purely algebraic and we will almost exclusively deal with finite dimensional complex Lie algebras.

First we shall give the basic definitions and first properties of Lie algebras. We will discuss many examples and concrete realizations. Then we shall study three important classes: the nilpotent, the solvable and the semisimple Lie algebras. The latter class will be the object for an intensive investigation. The highlight here will be a theorem (the main result of "the greatest mathematical paper of all times", see Math. Intel. 11 no. 3, 29-38) which classifies them in terms of their so-called root systems.

Throughout this structure theory we shall emphasize the representations of Lie algebras. The course will end by a closer look at some of the main features of representations of semisimple Lie algebras.

The course may be continued in several directions, e.g. representations of algebraic groups, quantum groups, Kac-Moody algebras or modular representations (where the complex field is replaced by a field of positive characteristic).

Prerequisites:

Algebra I

Literature:

J. E. Humphreys:,"Introduction to Lie Algebras and Representation Theory", Grad. Texts in Maths, Springer Verlag.