Appears in collection : Vyacheslav Futorny: Representations of Lie Algebras

The purpose of the mini-course is to give an introduction to the representation theory of Lie algebras, both finite and infinite dimensional: simple complex finite dimensional, Affine Kac-Moody algebras and Lie algebras of vector fields. We will discuss first the classification of simple finite dimensional complex Lie algebras and their finite dimensional representations, followed by the classical highest weight theory. Next we proceed to the theory of Kac-Moody algebras and focus on representations of Affine Lie algebras. In particular, we will discuss the free field realizations of these algebras and the origins of the vertex theory. Finally we describe classical and new results on the representations of Lie algebras of vector fields on algebraic varieties.