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

Abstract

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.