Category O and the basic ideas behind categorifications

In this course we will give an introduction into the so-called category O introduced originally by Bernstein, Gelfand and Gelfand. It is maybe the most important category arising in Lie theory and is the prototype for many constructions done in representation theory. In particular it is the first example of what is called a highest weight category and coined many notions used nowadays in modern representation theory.

We will introduce this category and in particular study hmological properties and the very crucial notion of translation functors. A good knowledge of Lie theory, in particular of Verma modules is assumed. More concretely it is expected that students know the material from either the course on semisimple Lie algebras or from Humphreys book on Introduction into Lie algebras and representation theory.

The second part of the course will then illustrate many basic concepts from a quite recent development, namely the idea of categorification. The main example will be provided by category O.

As concrete applications we will consider Khovanov homology which is a categorification of the Jones polynomial of knots and generalizations thereof.

We will also illustrate how categorifications can be used to obtain new results in representation theory, like finding decomposition numbers, establishing new equivalences of categories etc. These are all results developed in the last 15 years.

Rough outline of the course

Part I) Category O

- Basic definitions and the concept of translation functors,

- Highest weight categories

Part II Basics on Categorifications

- Different levels of catgeorifications: weak categorifications verus strong categorifications

- 2-categories and why they are usful

Part III Lie algebra actions on categories

- Basic examples sl_2-actions (these lead to the proof of the famous Broue-conceture for the symmetric groups which predicted non-trivial derived equivalences of categories of representations. In partucular it could be shown that a block depends (up to derived equivalence) on on the so-called defect

- generalizations to more general Lie algebras

- uniqueness of (tensor product) categorifications

References: For the first part of the course Humphreys' book on category O and semisimple Lie algebras is teh main reference. For the second part the references will be given in class. The article of Chuang and Rouquier on sl_2 categorifications and Khovanov's categorification of the Jones polynomial will be two  of the most important applications.