On non-counital Frobenius algebras

Abstract

A Frobenius algebra is a finite-dimensional algebra [Formula: see text] which comes equipped with a coassociative, counital comultiplication map [Formula: see text] that is an [Formula: see text]-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map [Formula: see text] as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.