Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

Higher Representation Theory and Categorification of Verma Modules

In Higher Representation Theory the usual basic structures of representation theory, like vector spaces and linear maps, are replaced by category theory analogs, like categories and functors. Opposite to vector spaces and linear maps, the world of categories is tremendously big, offering enough room for finding richer structures: for example, replacing linear maps by a functors always comes accompanied by a "higher structure" which is associated to natural transformations between them. This higher structure is invisible to traditional representation theory. Part 1: "What is higher representation theory and why should I care?" In the first part I will give an introduction to higher representation theory, with emphasis on higher representation theory of Lie algebras. In particular, I will describe a categorification of the finite-dimensional irreducible represenations of quantum sl2 using cohomologies of (finite dimensional) Grassmannians and partial flag varieties. Part 2: "DG-enhanced cyclotomic KLR algebras and categorification of Verma modules" In the second part I will present DG-enhanced versions of cyclotomic Khovanov-Lauda-Rouquier algebras and explain how to use them to categorify parabolic Verma modules for (symmetrizable) quantum Kac-Moody algebras.