This is a four hour lecture course to be given at Northeastern in the week of September 12-16.
Tuesday, Nov 1, 10am-12pm, Thursday, Nov 3, 3.45-5.45pm, Lake 509.
Title: Kac-Moody 2-categories and higher representation theory
Abstract: In the 1990s, several examples of categories equipped with an action of a Kac-Moody algebra g appeared in classical representation theory. These categories possess some exact endofunctors E_i and F_i, which induce linear maps on the Grothendieck group satisfying the Serre relations of the Lie algebra g. Moreover, there are natural transformations between these endofunctors which satisfy some higher relations, making the categories into 2-representations of the Kac-Moody 2-category associated to g.
These notions were formalized originally in the special case g = sl_2 by Chuang and Rouquier in 2004. They developed a powerful structure theory, which they used to prove a conjecture of Broue about existence of derived equivalences between blocks of symmetric groups. The general definition of Kac-Moody 2-category emerged in 2008, and is due to Rouquier and (independently) Khovanov and Lauda. Since then, the theory has had many other striking applications to classical representation theory.
In my first lecture, I will review the representation theory of the nil-Hecke algebra and some of the original results of Chuang and Rouquier in the sl_2 case, before defining the Kac-Moody 2-category in general. In the second lecture, I will focus on examples and applications, either to the representation theory of the symmetric group, or to the Bernstein-Gelfand-Gelfand category O.
Notes for lecture 1 (by Jose Simental).
Notes for lecture 2 (by Seth Shelley-Abrahamson).