Bimodules over uniformly oriented A n quivers with radical square zero

Abstract

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, we determine existing adjunctions between its $1$-morphisms, and we 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.