2-Verma modules (Q2065868)

This paper constructs a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using Khovanov-Lauda-Rouquier (KLR)-like diagrammatic algebras. The construction proposed in this paper arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. In particular, the usual categorification of integrable modules can be recovered from the proposed construction of parabolic Verma modules. The paper develops some basics of dg-2-representation theory, in particular, by introducing the notion of a dg-2-representation for a quantum Kac-Moody algebra. The results of this paper have very interesting applications in topology. For example, one can construct Khovanov-Rozansky's triply graded link homology using parabolic 2-Verma modules.