# Are local finite dimensional Hopf algebras symmetric?

Recall that a finite dimensional algebra $$A$$ over a field $$K$$is called a Frobenius algebra in case $$A \cong D(A)$$ as right modules, where $$D(A) \cong Hom_K(A,K)$$. In case $$A \cong D(A)$$ as bimodules, then $$A$$ is called a symmetric algebra. It is well known that a finite dimensional Hopf algebra is a Frobenius algebra.

Question: Is a finite dimensional local Hopf algebra a symmetric algebra?

Examples of local finite dimensional Hopf algebras are group algebras of $$p$$-groups and it was shown in When is the exterior algebra a Hopf algebra? that for the exterior algebra, it is indeed true that being a Hopf algebra implies that it is symmetric.

• Could you restate your question in English, please? – Bugs Bunny Dec 5 '18 at 11:07
• @BugsBunny I wrote a new formulation of the question. – Mare Dec 5 '18 at 11:16
• I think the answer is NO. By Oberst-Schneider symmetricity is equivalent unimodularity and $S^2$ being inner. Something like $u_q (n)$ at a root of unity should give a counterexample. – Bugs Bunny Dec 10 '18 at 14:13