Classification of non-commutative, non-cocommutative Hopf algebras

Question

1. Context
Apparently, Sweedler's Hopf algebra (presented in 1969) was the first known example of a non-commutative, non-cocommutative Hopf algebra.

More generally, the $N^2$-dimensional Taft-Hopf algebra $H_{N^{2}}$ (introduced in 1971) yields a non-commutative, non-cocommutative Hopf algebra for every positive integer $N \neq 1$ (and an appropriate field). Sweedler's Hopf algebra is simply Taft's Hopf algebra $H_4$ over a field $\mathbb k$ (with $\zeta =-1$ and $char(\mathbb k) \neq 2$).

These examples are more or less from the 1970s. I am wondering what the current state of affairs is.

2. Questions

• What other non-commutative, non-cocommutative Hopf algebras are known?

• I skimmed the wikipedia article on the Pareigis Hopf algebra. It seems that it is a further example of a non-commutative and non-cocommutative Hopf algebra. What is its dimension?

• Is there a (complete or partial) classification (up to isomorphism)? What about the subclass of finite-dimensional Hopf algebras?

Answer 1

Where I would look for examples of non-commutative and non-cocommutative Hopf algebras is the world of quantum groups, where you will find many such examples (that also carry a *-involution)

For example,

1. The algebra of functions on the Kac-Paljutkin quantum group (dimension eight).
2. The algebra of functions on a Sekine quantum group (dimension $2n^2$ for a natural number parameter $n$).
3. The algebra of regular functions on Wang's quantum permutation groups, equal to $F(S_N)$ for $N\leq 3$ but infinite dimensional for $N\geq 4$.

To see more examples see the references here.

I can't answer your second question.

Your third question, perhaps look at the papers of Andruskiewitsch.