Infinite Dimensional Lie Algebras

An Introduction

1. Basic definitions.
2. The invariant bilinear form and the generalized Casimir operator.
3. Integrable representations and the Weyl group of a Kac-Moody algebra.
4. Some properties of generalized Cartan matrices.
5. Real and imaginary roots.
6. Affine Lie algebras: the normalized invariant bilinear form, the root system and the Weyl group.
7. Affine Lie algebras: the realization (case k = 1).
8. Affine Lie algebras: the realization (case k = 2 or 3). Application to the classification of finite order automorphisms.
9. Highest weight modules over the Lie algebra g(A).
10. Integrable highest weight modules: the character formula.
11. Integrable highest weight modules: the weight system, the contravariant Hermitian form and the restriction problem.
12. Integrable highest weight modules over affine Lie algebras. Application to ?-function identities.
13. Affine Lie algebras, theta functions and modular forms.
14. The principal realization of the basic representation. Application to the KdV-type hierarchies of non-linear partial differential equations.
Index of notations and definitions.
References.