Characterisation and applications of $\mathbb{k}$-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 $\mathbb{k}$-split in the sense that they factor (inside the tensor category of bimodules) over $\mathbb{k}$-vector spaces. As one application, we show that any simple $2$-category has a faithful $2$-representation inside the $2$-category of $\mathbb{k}$-split bimodules. As another application, we classify simple transitive $2$-representations of the $2$-category of projective bimodules over the algebra $\mathbb{k}[x,y]/(x^2,y^2,xy)$.

References in corpus (2)

• 2-Kac-Moody algebras
• Simple transitive 2-representations via (co)algebra 1-morphisms

Cited by in corpus (5)

• Bimodules over uniformly oriented A_n quivers with radical square zero
• Analogues of centralizer subalgebras for fiat 2-categories and their 2-representations
• Pyramids and 2-representations
• Sub-bimodules of the identity bimodule for cyclic quivers
• Cell 2-Representations and Categorification at Prime Roots of Unity