We give a geometric categorification of the Verma modules M (λ) for quantum sl2. ContentsFrom the work of Lauda in [34,35] adjusted to our context it follows that for each n 0 there is an action of the nilHecke algebra NH n on F n and on E n . As a matter of fact, there is an enlargement of NH n , which we denote as A n , acting on F n and E n , also admitting a nice diagrammatic description (see § 1.1.6 for a sketch).We define M as the direct sum of all the categories M k and functors F, E and Q in the obvious way. One of the main results in this paper is the following. † All our functors are, in fact, superfunctors which we tend to see as functors between categories endowed with a Z/2Z-action, whence the use of the terminology functor. ‡ We thank Aaron Lauda for explaining this to us.

We construct an odd version of Khovanov's arc algebra H n . Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the (n, n)-Springer variety. We also prove that the odd arc algebra can be twisted into an associative algebra.Lemma 5.11. The geometric realization of N(G n ) is a simplex of dimension C n − 1, for C n the n th Catalan number: |N(G n )| ≃ ∆ (Cn−1) .

No abstract

We construct a categorification of parabolic Verma modules for symmetrizable Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac–Moody algebras, and in particular of parabolic 2-Verma modules.

We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac-Moody algebras, and in particular of parabolic 2-Verma module.