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

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

“…Recently, the fact that simple transitive 2-representations of C Λ are exhausted by cell 2-representations was proved in full generality (that is for arbitrary Λ) in [MMZ2]. Some intermediate results appeared in [MZ1,Zi2,MMZ1].…”

Pyramids and 2-representations

“…The weak Jordan-Hölder theory developed in [MM5] motivates the study of so-called simple transitive 2-representations which are suitable 2-analogues of simple modules. It turns out that, in many cases, simple transitive 2-representations can be explicitly classified, see [MM5,MM6,MaMa,KMMZ,MMMT,MMZ,MT,MZ1,MZ2,Zh,Zi1,Zi3] for the results and [Maz2] for a detailed survey on the subject. In many, but not all, cases, simple transitive 2-representations are exhausted by so-called cell 2-representations defined already in [MM1].…”

“…With this result in hand, it is very natural to consider 2-categories of projective bimodules for arbitrary finite dimensional algebras as a basic family of simple 2-categories. Outside the self-injective case, a number of examples were studied in [MZ1,MZ2,MMZ,Zi3] and in all cases, using rather different arguments, it was shown that simple transitive 2-representations are exhausted by cell 2-representations.…”

“…The 2-category D A loses some of the symmetries of C A but compensates for this loss by possessing many pairs of adjoint 1-morphisms. With the machinery developed in [KiM,KMMZ,MZ1,MMZ] and some tricks using matrix computations, we prove in Theorem 9 that simple transitive 2-representations of D A are exhausted by cell 2-representations.…”

“…Let X and Y be the B-B-bimodules corresponding to the actions of M(F) and M(G), respectively. By [KMMZ,Theorem 11(i)] (which does not make use of the fiatness assumption), both X and Y have the property that the B-modules X ⊗ B L and Y ⊗ B L are projective, for any B-module L. Therefore, by [MMZ,Theorem 1], both X and Y are of the form…”