Home Page of Alistair Savage | uOttawa Department of Mathematics and Statistics | CRM Thematic Semester |

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* (part of the CRM thematic semester New Directions in Lie Theory). 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.

Students should have a solid background in groups, rings, and modules covered, for instance, in typical first year graduate courses in algebra. A basic understanding of the fundamentals of category theory (for example, the first chapter of Categories for the working mathematician by Mac Lane) will also be assumed.

The following references are more or less introductory and involve some of the topics we will discuss in the course.

- Wei Lu and Aaron McBride, Algebraic structures on Grothendieck groups, 2013.
- Alexander Kleshchev, Linear and projective representations of symmetric groups, Cambridge University Press, 2009.
- Volodymyr Mazorchuk, Lectures on algebraic categorification, European Mathematical Society, 2012.
- Seok-Jin Kang, An elementary introduction to categorical representation theory, 2013.
- Aaron Lauda, An introduction to diagrammatic algebra and categorified quantum sl
_{2}, 2012.

The references below are more advanced and are for students interested in learning more about categorification and its applications.

- Jonathan Brundan, Quiver Hecke algebras and categorification, 2013.
- Joel Kamnitzer, Categorification of Lie algebras [d'apès Rouquier, Khovanov–Lauda], 2013.
- Mikhail Khovanov, Volodymyr Mazorchuk, and Catharina Stroppel, A brief review of abelian categorification, 2009.

Lecture notes are posted below.

A more detailed syllabus can be found here.