Convolution Products and R-Matrices for KLR Algebras of Type $B$

TL;DR

This paper develops convolution products and R-matrices for modules over VV algebras of type B, exploring their properties and potential links to quantum cluster algebra structures.

Contribution

It introduces convolution products and R-matrices for VV algebra modules of type B, advancing the categorification and algebraic understanding of these structures.

Findings

Defined convolution products for VV algebra modules

Constructed R-matrices satisfying Yang-Baxter equation

Explored properties of R-matrices related to simple modules

Abstract

In this paper we define and study convolution products for modules over certain families of VV algebras. We go on to study morphisms between these products which yield solutions to the Yang-Baxter equation so that in fact these morphisms are $R$-matrices. We study the properties that these $R$-matrices have with respect to simple modules with the hope that this is a first step towards determining the existence of a (quantum) cluster algebra structure on a natural quotient of $\mathcal{B}_\theta(\mathfrak{g})$, the $\mathbb{Q}(v)$-algebra defined by Enomoto and Kashiwara, which the VV algebras categorify.