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

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 $\mathfrak{s}{\mathfrak{l}}_2$. 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.

arXiv commentv1, 81 pages (52 without Appendices and references)

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