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.Comment: v2, substantial revision, introduction of a notion of dg-2-representation, 65p

A remark on Rasmussen's invariant of knots

We show that Rasmussen's invariant of knots, which is derived from Lee's variant of Khovanov homology, is equal to an analogous invariant derived from certain other filtered link homologies.Comment: There are two errors in the proof of Proposition 3.2. in this paper. These are indicated in an erratum added at the beginning. As a consequence the proof of Proposition 3.2 no longer holds and the proof of Theorem 4.2, which relies on it, is no longer vali

Sl(N) link homology using foams and the Kapustin-Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.Comment: The Kapustin-Li formula has been corrected for facets with non-zero genus and its normalization and some signs have been changed accordingly. 43 pages, lots of figure

Super qq-Howe duality and web categories

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.Comment: 38 pages, many colored figures, extra section containing new results, added suggestions of two referees, comments welcom

Tensor product categorifications, Verma modules and the blob 2-category

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of $\mathfrak{sl}_2$. From the above we derive a categorification of the blob algebra.Comment: v2, 88 pages, reviewed versio