# Bimodules over uniformly oriented A_n quivers with radical square zero

## TL;DR

This paper classifies bimodule categories over certain finite-dimensional algebras derived from uniformly oriented A_n quivers with radical square zero, analyzing their structure and representation theory.

## Contribution

It provides a detailed description of the cell structure, adjunctions, and simple transitive 2-representations of the tensor category of bimodules over these algebras.

## Key findings

- Classification of connected finite-dimensional algebras with finitely many bimodule classes
- Description of the cell structure of the bimodule tensor category
- Proof that all simple transitive 2-representations are cell 2-representations

## Abstract

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-representation of the $2$-category of projective bimodules is equivalent to a cell $2$-representation.