J-equivalence for associative algebras

Abstract

We study the combinatorics of an analogue of Green’s $${\mathcal {J}}$$ J -relation (a.k.a. the two-sided relation) for the bicategory of finite-dimensional bimodules over finite-dimensional associative algebras over a fixed field. In particular, we provide a number of exameples for both $${\mathcal {J}}$$ J -equivalent and inequivalent but $${\mathcal {J}}$$ J -comparabe algebras.