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I'm looking for a finite-dimensional Hopf algebra (over any field) that is unimodular, has unimodular dual, but is not involutory. Is there such a thing?

Here's what I know:

  • By Radford's formula, the antipode $S$ must have order 4.

  • Suzuki has constructed unimodular Hopf algebras over any field that are not involutory (and where $S^2$ is not even inner). My calculations so far suggest that these are not counimodular, but I still need to double-check.

  • It is well-known that semisimplicity implies unimodularity. Therefore finding a cosemisimple unimodular Hopf algebra that is not involutory would be even stronger (or dually, a semisimple counimodular one).

  • A result of Etingof and Gelaki is that "semisimple cosemisimple" implies involutory, so this combination would be too strong.

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It turns out that an explicit example of such a Hopf algebra already appears at the end of Radford's seminal paper The Order of the Antipode of a Finite Dimensional Hopf Algebra is Finite as Example 2.

The example is a bit too complicated to reproduce here, so let me just note that it is 8-dimensional, can be defined and has the desired properties over any field, and happens to be isomorphic to its opposite.

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