On non-counital Frobenius algebras

TL;DR

This paper investigates non-counital comultiplication structures in generalizations of Frobenius algebras, especially self-injective algebras, including weak Hopf algebras, and conjectures their broader applicability.

Contribution

It demonstrates that many self-injective algebras possess non-counital comultiplication maps, extending the concept beyond classical Frobenius algebras and proposing a general conjecture.

Findings

Large classes of self-injective algebras have nonzero non-counital comultiplication maps.

Finite-dimensional weak Hopf algebras admit such comultiplicative structures.

Conjecture that all self-injective algebras may have similar comultiplication maps.

Abstract

A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped with a coassociative, counital comultiplication map $\Delta$ that is an $A$-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 $\Delta$ as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.