J-equivalence for associative algebras

Abstract We study the combinatorics of an analogue of Green’s $${\mathcal {J}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -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 $${\mathcal {J}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -equivalent and inequivalent but $${\mathcal {J}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>J</mml:mi> </mml:math> -comparabe algebras.