# Explicit examples of finite dimensional, involutive Hopf algebras with traceless antipode?

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In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $$H$$ where the antipode $$S:H \to H$$ is traceless.

Unfortunately, the method of proof in [1] seems rather inexplicit to me. My understanding is that the authors use the fact that every fusion category $$\mathcal{C}$$ satisfying some conditions is the representation category of some Hopf algebra. Then they convert the trace condition $$\text{Tr}(S) = 0$$ into a condition on the fusion category $$\mathcal{C}$$ and show that they can easily construct categories satisfying this new condition, in addition to the conditions necessary for $$\mathcal{C}$$ to be a representation category of some $$H$$.

The paper [1] is eight years old now, so I'm hoping that some more explicit examples might be in the literature at this point. So my question is:

Question: What are some explicit examples of a finite dimensional, involutory Hopf algebra (over a field $$k$$ of characteristic $$0$$) where the antipode is traceless? By explicit, I mean given by a generator/relation presentation or as some structure tensors in a specific basis.

Here I weakened the hypothesis to assume that $$H$$ is just involutory, i.e. that $$S^2 = \text{Id}$$. A theorem of Larson and Radford [2] says that semisimple Hopf algebras are involutory when $$\text{char}(k) = 0$$.

Alternatively, I would appreciate some advice on what I need to know in order to extract an explicit example from [1], if that is what's necessary.