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.