Affine Kac-Moody Lie algebras (Аффинные алгебры Каца-Муди)


The theory of Lie groups and Lie algebras is one of the central areas of modern mathematics. It has various interrelations
with algebraic geometry, combinatorics, theory of symmetric functions, integrable systems, classical and quantum field theories.
Lie groups and Lie algebras usually show up as the sets of symmetries of objects of a theory.
For example, affine Kac-Moody algebras turn out to be very important
for the description of many quantum field theories showing up as symmetries of the spaces of states.
These Lie algebras also play an important role in the theory of integrable systems and in algebraic geometry.
Our goal is to give an introduction to the theory of affine Kac-Moody algebras.
We will describe main definitions, constructions and applications of the theory.

Prerequisites: basic Lie theory.

Curriculum:

Finite-dimensional simple Lie algebras: Cartan subalgebras, root vectors: Weyl group.
Finite-dimensional simple Lie algebras: Verma modules, irreducible representations, characters.
Generalized Cartan matrices, Kac-Moody Lie algebras - definitions and first properties..
Finite, Affine and Indefinite Kac-Moody Lie algebras.
Classification of affine Kac-Moody Lie algebras.
Weyl group and invariant bilinear form.
Real and imaginary roots.
Canonical central element and imaginary roots for affine Lie algebras.
Center and the Weyl group for affine Lie algebras.
Weyl group for affine Lie algebras and alcoves.
Affine walls and alcoves.
Loop algebras and central extensions.

Textbooks:
V. Kac, Infinite dimensional Lie algebras, Cambridge University Press.
R.Carter, Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics .