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

TL;DR

This paper constructs a categorification of tensor products of Verma and integrable modules for quantum sl2 using derived categories of dg KLRW algebras, advancing the understanding of braid group actions in representation theory.

Contribution

It introduces a categorification framework for all tensor products of Verma and integrable modules for quantum sl2 via dg KLRW algebras, generalizing previous algebraic structures.

Findings

Computed bases for dgKLRW algebras using rewriting methods

Established a categorification of tensor products for quantum sl2 modules

Developed techniques to handle braid-like isotopy in algebraic structures

Abstract

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $\mathfrak{sl_2}$. The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using rewriting methods modulo braid-like isotopy, which we develop in an Appendix.