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

Abstract

We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl 2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are twodimensional. This action intertwines with the categorical action of sl 2 . From the above we derive a categorification of the blob algebra.

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

“…They also showed in [37] that their construction is related to Khovanov-Rozansky triply-graded link homology [26]. Moreover, in a collaboration [27] with Lacabanne, they gave a categorification of the tensor product of a Verma module with multiple integrable modules for quantum sl 2 . They also showed that their construction yields a categorification of the blob algebra of Martin-Saleur [32], which allow the construction of invariants of tangles in the annulus.…”

“…Before defining the dgKLRW algebras, we fix some conventions, and we recall some common facts about dg-structures (classical references for this are [19] and [44], see also [34, Appendix A] for a short survey oriented towards categorification), and about rewriting methods. Since we use the same conventions as in [27], a part of this section is almost identical to [27, §3.1 and Appendix B].…”

“…Inspired by the KLRW algebra in [47, §4] (called "tensor product algebra" in the reference), which we think of as associated to a string of dominant integral g-weights, and generalizing the dg-enhanced KLRW algebra in [27], which we think of as associated to a generic weight β and a string of dominant integral g-weights, we introduce a dgKLRW (dg-)algebra associated to a string of weights that can each either be generic or integral. Definition 4.1.…”

“…Proof. If |µ| P N, then it is [27,Proposition 4.3]. Suppose |µ| R N. Since h µ p1 ρ q is symmetric w.r.t.…”

“…This is the first part of a series of two papers aiming to construct and study more general tensor products of Verma and integrable modules. In this first part, we propose a categorification of any such tensor product for quantum sl 2 using dgKLRW algebras, generalizing the construction in [47] and in [27]. In a second part in preparation [13], we construct a categorical braid group action lifting the action of the R-matrix.…”