$2$-categories of symmetric bimodules and their $2$-representations

Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang

Vol. 306 (2020), No. 2, 645–678

DOI: 10.2140/pjm.2020.306.645

Abstract

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under which condition the resulting $2$-category is fiat (in the sense of Mazorchuk and Miemietz (2011)) and classify simple transitive $2$-representations of this $2$-category (under some mild technical assumption). We also study several classes of examples in detail.

Keywords

finite abelian group, symmetric bimodule, 2-category, simple transitive 2-representation

Mathematical Subject Classification 2010

Primary: 18D05

Milestones

Received: 5 May 2019

Revised: 13 February 2020

Accepted: 13 February 2020

Published: 13 July 2020