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It was shown by Abrams (see https://www.sciencedirect.com/science/article/pii/S0021869399979012 ) that every Frobenius algebra has a canonical coalgebra structure.

Question 1: Has it been studied when the Frobenius algebra together with the coalgebra structure is a Hopf algebra?

It seems that this can happen nearly never but I am not so experienced with Hopf algebras. The question sounds also rather natural so probably it has been studied already or has a trivial answer (that there are no non-trivial cases?)

Question 2: Is there a program that can test whether a given Frobenius algebra (given by quiver and relations) is a Hopf algebra ? (for example finding all possible coalgebra structures and testing for the Hopf algebra conditions).

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  • $\begingroup$ Concerning question 2, do you know the work of C. Cibils and M. Rosso? I don't hace the precise reference, but look for them on Google together with Hopf quivers, or Hopf path algebras/ Hopf path coalgebras. They look when a path algebra/coalgebra is Hopf. I don't remember if they consider relations. There are very few quivers that works $\endgroup$
    – Marco Farinati
    Commented Jun 19, 2020 at 18:02

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The compatibility conditions on the product/coproduct for Frobenius algebras and Hopf algebras are very different. It's almost never the case that the same coalgebra/algebra satisfies both the Frobenius axioms and the Hopf algebra axioms. (Probably it's not hard to prove a result in that direction, but I don't know it off the top of my head. The trivial example of just the base field will satisfy both.)

One weird complication is that finite dimensional Hopf algebras always have a Frobenius algebra structure, but the coproduct for the Hopf algebra structure and the coproduct for the Frobenius structure are different! In particular, the counit for the Frobenius structure is given by an integral, not by the counit.

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In case $A$ is a non-semisimple quiver algebra, the trace $\lambda$ of the Frobenius structure satisfies $\lambda(1)=0$ (since it is essentially given so that it maps only elements of the socle of the algebra to non-zero elements) and thus it can not be an algebra map. Thus in the case of non-semisimple quiver algebras, this coalgebra structure indeed never gives rise to a Hopf algebra structure.

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