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

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.