Vol. 306, No. 2, 2020

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

Volodymyr Mazorchuk, Vanessa Miemietz and Xiaoting Zhang

Vol. 306 (2020), No. 2, 645–678
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
Authors
Volodymyr Mazorchuk
Department of Mathematics
Uppsala University
Uppsala
Sweden
Vanessa Miemietz
School of Mathematics
University of East Anglia
Norwich
United Kingdom
Xiaoting Zhang
Academy for Multidisciplinary Studies
Capital Normal University
Beijing
China