Are there small examples of non-pivotal finite tensor categories?

Question

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with.

Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category whose underlying category is equivalent to the category of finite dimensional modules over a finite dimensional algebra. A tensor category is pivotal if there's an isomorphism of tensor functors between the identity functor and the double dual functor.

The example I was able to find in the literature is the category of representations of a 72-dimensional Hopf algebra in Remark 2.11 of Andruskiewitsch-Angiono-Iglesias-Torrecillas-Vay. (In fact, they give three such examples.) But I was hoping for a smaller example.

Bonus question: What I would really like is an example of a category where not only is the double dual functor nontrivial, but there's no invertible object X such that the double dual is isomorphic as a tensor functor to conjugation by X. (Note that the 72 dimensional Hopf algebras mentioned above do not give counterexamples, because their duals are pivotal.)

Easier question: I'd also love to hear about any other examples of non-pivotal finite tensor categories beyond the three from AAITV, even if they're not smaller.

Answer 1

Monoidal categories $\mathcal{C}$ which admit a monoidal natural isomorphism $\Phi_X:X^{**} \rightarrow \beta\otimes X\otimes \beta^*$ are called $\textit{quasi-pivotal}$ . For $\mathcal{C}=$Rep$(H)$ with $H$ a finite dimensional Hopf algebra, quasi-pivotal structures on $\mathcal{C}$ are in bijection with pairs $(l,b)$ where $l$, $b$ are group-like elements in $H$, $H^*$ respectively satisfying $$S^2(h)=b(h_3)\;b^{-1}(h_1)\; l\;h_2 \; l^{-1} \;\;\;\; \text{for all}\;\; h\in H.$$
Such a pair $(l,b)$ is called a $\textit{pair in involution}$. The paper Generalized Taft algebras and pairs in involution, constructs a family of Hopf algebras that do not admit a pair in involution, thereby providing an answer to the bonus question.

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"Book Hopf algebras" provide an example of Hopf algebras that admit a quasi-pivotal structure but don't always admit a pivotal structure. See this paper for further details.

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Furthermore, by a result of Shimizu, a quasi-pivotal structure on $\mathcal{C}$ yields a pivotal structure on $\mathcal{Z}(\mathcal{C})$. Thus, the Drinfeld doubles $D(H)$ of the Hopf algebras admitting a pair in involution (discussed above) provide examples of pivotal Hopf algebras.