Higher representation theory
We have introduced a 2-category associated with a Kac-Moody algebra (the type A case goes back to joint work with Joe Chuang and a close version of the positive half has been introduced independently by Khovanov and Lauda). We will discuss the 2-representation theory, ie, actions of this 2-category on categories (additive, abelian, triangulated, dg...). We will present a unicity result for simple integrable 2-representations and Jordan-Holder series. We will explain the realisation of simple 2-representations as categories of sheaves on quiver varieties and deduce the description of classes of indecomposable projective modules as canonical basis elements.