Long dimodules and quasitriangular weak Hopf monoids

Abstract

In this paper, we prove that for any pair of weak Hopf monoids H and B in a symmetric monoidal category where every idempotent morphism splits, the category of H-B-Long dimodules HBLong is monoidal. Moreover, if H is quasitriangular and B coquasitriangular, we also prove that HBLong is braided. As a consequence of this result, we obtain that if H is triangular and B cotriangular, HBLong is an example of a symmetric monoidal category.