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: In case a Hopf algebra $A$ is a local finite dimensional algebra, is $A$ even symmetric?

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.

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.

I'm searching for (a class of) examples of Hopf algebras , which have the following properties:

they should be finite dimensional

they should not be semisimple

they should be local

they should not be symmetric (or if this is not possible, weaken the condition to:

they should not be Morita equivalent to local blocks of group algebras)

Note that a finite dimensional Hopf algebra is always selfinjective and a class of examples of local selfinjective nonsemisimple finite dimensional Hopf algebras are the group algebras of p-groups over a field of characteristic p ( but they are 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: In case a Hopf algebra $A$ is a local finite dimensional algebra, is $A$ even symmetric?

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.