When weak Hopf algebras are Frobenius

TL;DR

This paper explores conditions under which weak Hopf algebras are Frobenius, revealing that uniform matrix block dimensions in the base algebra ensure Frobenius property, and extends the concept to a categorical framework.

Contribution

It establishes criteria for weak Hopf algebras to be Frobenius based on base algebra structure and generalizes the Frobenius property to a categorical setting.

Findings

Frobenius property holds if base algebra's matrix blocks are uniform in size.

Counterexamples exist when base algebra lacks uniform matrix block dimensions.

Categorical analogue of Frobenius property for noncoassociative weak Hopf algebras.

Abstract

We investigate when a weak Hopf algebra H is Frobenius; we show this is not always true, but it is true if the semisimple base algebra A has all its matrix blocks of the same dimension. However, if A is a semisimple algebra not having this property, there is a weak Hopf algebra H with base A which is not Frobenius (and consequently, it is not Frobenius "over" A either). We give, moreover, a categorical counterpart of the result that a Hopf algebra is a Frobenius algebra for a noncoassociative generalization of weak Hopf algebra.