Characterisation and applications of $\Bbbk$-split bimodules

Abstract

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are $\Bbbk $-split in the sense that they factor (inside the tensor category of bimodules) over $\Bbbk $-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\Bbbk $-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\Bbbk [x,y]/(x^2,y^2,xy)$.