$ required {AMScd} $

In the newspaper [1]It is shown that there are finally three-dimensional Hopf algebras $ H $ where the antipode $ S: H to H $ is without a trace.

Unfortunately, the proof method is in [1] seems to me rather inexplicable. I understand that the authors use the fact that every fusion category

$ mathcal {C} $ Fulfilling some conditions is the representation category of some Hopf algebra. Then you convert the trace condition $ text {Tr} (S) = 0 $ in a condition of the fusion category $ mathcal {C} $ and show that they can easily create categories that meet this new condition, in addition to the required conditions $ mathcal {C} $ be a representation category $ H $,

The paper [1] is now eight years old, so I hope that some more explicit examples in the literature could be at this point. My question is:

Question:What are some explicit examples of a finite, inclusive Hopf algebra (across a field)? $ k $ of characteristic $ 0 $) where the antipode is without a trace? By explicit I mean given by a generator / relationship representation or as some structure tensors on a certain basis.

Here I weakened the hypothesis to assume it $ H $ is only indicative, i. $ S ^ 2 = text {Id} $, A sentence by Larson and Radford [2] says that semisimple Hopf algebras are involved, though $ text {char} (k) = 0 $,

Alternatively, I would appreciate some advice that I need to know to get an explicit example [1]if necessary.