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

Question

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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.

Answer 1

Unfortunately I can only address the alternative (and 4 years too late >_<).

The result of Tambara that Shimizu uses is that such a category admits a fiber functor. For your purposes, you would need to have an explicit formulation of the fiber functor, and for this you can use Tambara's classification result Thm 3.5. For simplicity, let us choose the smallest such example, where $r=1$ and $A=(\mathbb Z/4\mathbb Z)^2$.

Step 1) Realize the category using the construction of Tambara and Yamagami in Thm Def 3.1, with inputs $A$, $\chi$ and $\tau$ as Shimizu indicates.

Step 2) Translate the $(\sigma,\rho)$ that Shimizu writes down into monoidal structure maps $$t_{X,Y}:F(X)\otimes F(Y)\to F(X\otimes Y)\,.$$ This is a bit of a pain, because the recipe that Tambara provides must be read in reverse in order to write down the appropriate coefficients. The coefficients that you are hoping to write down are listed as Equations (1)-(8) on page 40.

As a linear functor, the desired fiber functor is the forgetful functor that takes every object to a vector space of the appropriate dimension. The objects corresponding to elements of $A$ will have dimension 1, whereas the additional object $m$ will have dimension $4$.

Step 3) Now you have a fusion category $\mathcal C=\mathcal C(A,\chi,\tau)$ and a fiber functor $(F,t):\mathcal C\to\text{Vect}$, and you can use Tannaka-Krein duality to give the desired Hopf algebra $$H:=\text{End}(F)$$ the algebra of natural transformations of $F$. For more on how the structure maps $\{t_{X,Y}\}_{X,Y\in \mathcal C}$ are used to establish the Hopf algebra structure, see Sec 5.2-5.3 of the Tensor Categories book by Etingof, Nikshych, Gelaki, and Ostrik.

In the end, using this choice of $A=(\mathbb Z/4\mathbb Z)^2$, this Hopf algebra should have dimension 32.