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I wish to find an example of a semisimple (hence finite dimensional) Hopf algebra $H$ with the following properties:

  • $H$ is nontrivial (i.e. not a group algebra or the dual of one);
  • The category of representations of $H$ is a monoidal category which is not symmetric (i.e. the Grothendieck ring is noncommutative);
  • There exists a two-dimensional representation $V$ which is inner-faithful (i.e. the only Hopf ideal which annihilates $V$ is the zero ideal).

I've checked various examples of Hopf algebras of low dimension and haven't been able to find an example with these properties. Unless I'm mistaken, the nontrivial examples in dimensions 8 and 12 don't have the desired properties, nor do the sixteen examples in dimension 16 which were classified by Kashina.

Seeking to construct an example with the desired properties, I have also considered abelian extensions of the form $(kG)^* \# \hspace{1pt} kL$ as in this paper. With a judicious choice of $G$ and $L$, one can ensure that the Grothendieck ring of such a Hopf algebra is noncommutative, but none of the examples I have checked have a two-dimensional inner-faithful representation.

Of course, it may also be the case that a Hopf algebra with these properties cannot exist, but I don't see why this should be the case.

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