This mini-course will serve to introduce students to the new and exciting field of categorification. Its goal is to prepare students for the workshop Geometric representation theory and categorification. The course will begin with a very brief review of the representation theory of associative algebras, before introducing the concept of weak categorification with some simple examples. In the second week, more sophisticated examples of categorification will be presented, including a categorification of the polynomial representation of the Weyl group and the Fock space representation of the Heisenberg algebra. The course will conclude with a discussion of more advanced ideas in categorification, such as strong categorification and the categorification of quantum groups.

While many universities offer a first course on Lie algebras, a second course on Lie algebras is usually not part of the graduate program. This mini-course will present the basic structure theory of Kac-Moody algebras with a special emphasis on affine Lie algebras (as found, for instance, in the first eight chapters of the book Infinite- dimensional Lie algebras by Kac). Some related non-Kac-Moody algebras, such as toroidal algebras, will also be presented. This course will prepare students for the mini-course “Representation theory of semisimple, affine and Kac-Moody algebras” in the second winter school (February 24 - March 7, 2014).