Mini-course “Representations of simple finite dimensional and affine Lie algebras”

This mini-course will focus on Representation Theory of simple complex finite dimensional and affine Lie algebras. We will start with the introduction to the theory of infinite dimensional weight representations. Though the classification of irreducible representations is still open beyond the sl(2) case in general, in some subcategories this problem can be managed successfully. In particular, we will discuss the theory of Gelfand-Tsetlin modules and their generalizations in the finite dimensional case. A comparison will be given with the representations of affine Kac-Moody algebras. We will conclude with the construction of free field realizations of these algebras and discuss open problems.