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As far as I know, low-dimensional semisimple Hopf algebras are classified (along with non-semisimple ones) up to dimension 60, with the first example of a semisimple Hopf algebra not coming from a finite group in dimension 8.

How well are low-dimensional Hopf superalgebras, that is, $\mathbb{Z}_2$-graded Hopf algebras or Hopf algebra objects internal to $\mathbb{Z}_2-\operatorname{Vect}$ understood? Up to which dimension are they classified? Are there interesting semisimple ones? Has someone worked out the representations? I could find an article on the classification of finite dimensional ones up to dimension 4, but it didn't mention semisimplicity or higher dimensional semisimple examples.

Generally, how well are low-dimensional semisimple braided Hopf algebras (internal to a braided category) understood?

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