# An Introduction to Affine Kac-Moody Algebras

## Master Class, 16 to 20 October 2006 by David Hernandez (CNRS, France)

**Abstract:**
Affine Kac-Moody algebras are
infinite dimensional
analogs of semi-simple Lie algebras and have a
central role
in both Mathematics and Mathematical Physics (Conformal Field
Theory).

In these lectures we will first explain how
appears naturally as a central extension
of the loop algebra of a semi-simple Lie algebra
. Then it is possible to define a system of
Chevalley generators which gives a unified point of view on
and .
The representation theory of is very
rich.

We study two classes of representations:

- the category O of representations: for example it
contains simple highest weight representations (they are not
finite dimensional except a few ones).

- the category of finite dimensional representations : for
example it contains representations obtained by evaluation
from finite dimensional
representations of .
has a central element which allows to
define the
level of a simple representation.

Then we will study more
advanced topics as the fusion product inside the category of
a given level, the critical level and applications to
Knizhnik-Zamolodchikov equations.

Professor David Hernandez will be visiting CTQM from 12 to 29 October 2006.

### Schedule

The schedule for the Master Class is as follows:

Week 42, Auditorium D4, Bldg. 1.531, 2nd floor

Monday 16 October to Friday 20 October:

10:00-10:45 Lecture

10:45-11:15 Coffee break

11:15-12:00 Lecture

### List of Participants

### Master Class Notes