How well is the classification of low-dimensional semisimple Hopf superalgebras (or braided Hopf algebras) understood?

Question

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?

Answer 1

If H is a Hopf super algebra then $H\#k[\mathbb Z_2]$ is an ordinary Hopf algebra. So, with some work, one should get the classification up to dim 30 from the classification of ordinary Hopf algebras up to dim 60

Answer 2

The OP asks more than one different things: the classification of fin dim, semisimple (or not), braided Hopf algebras is still a wide open area (up to my knowledge of course).

The classification of finite dimensional hopf superalgebras (the super- here is to be understood as the simplest case of braided) is another thing. It is still open in general, but a particular case of it has been solved long ago; this has to do with all the finite dimensional (super-)cocommutative hopf superalgebras. These have been classified by means of the following:

Let $\mathcal{H}$ be a finite dimensional, super-cocommutative, hopf superalgebra over an algebraically closed field $k$ of characteristic zero. Then $$\mathcal{H}\cong k[G(\mathcal{H})]\ltimes\bigwedge V$$ where $V$ is the space of primitive elements of $\mathcal{H}$ (regarded as an odd vector space), $\bigwedge V$ is its exterior algebra and $k[G(\mathcal{H})]$ is the group algebra of the the finite group $G(\mathcal{H})$ of the grouplikes of $\mathcal{H}$. In other words, $\mathcal{H}$ is a supergroup algebra.

Remarks:

1. The above proposition generalizes the corresponding classification for the ungraded case (see for example: About the classification of commutative and of cocommutative, fin. dim. Hopf algebras)

2. The above proposition, can be seen as a corollary (in the fin dim case) of the super-version of the Cartier-Constant-Milnor-Moore classification theorem, which is actually a classification result of super-cocommutative hopf superalgebras:

   Let $\mathcal{H}$ be a super-cocommutative hopf superalgebra over an algebraically closed field $k$ of char zero. Then we have the hopf superalgebra isomorphism: $$ \mathcal{H}\cong k[G(\mathcal{H})]\ltimes_{\pi} U\big(P(\mathcal{H})\big) $$ where, $k[G(\mathcal{H})]$ is the group algebra of the the group $G(\mathcal{H})$ of the grouplikes of $\mathcal{H}$, $U\big(P(\mathcal{H})\big)$ is the universal enveloping algebra of the lie superalgebra $P(\mathcal{H})$ of the primitive elements of $\mathcal{H}$ and the smash product $\ltimes_{\pi}$ is with respect to the representation of $G(\mathcal{H})$ on $P(\mathcal{H})$ determined by: $\pi:G\to Aut(P)$, $\pi(g)x=gxg^{-1}$, for all $g\in G$, $x\in P$.

   Furthermore, the hopf superalgebra $\mathcal{H}$ is finite dimensional if and only if $G(\mathcal{H})$ is a finite group and $P(\mathcal{H})$ is finite dimensional and purely odd (thus: an abelian lie superalgebra).

   For more details, see: theorem 3.3, p.224, B.Kostant, "Graded manifolds, graded Lie theory and prequantization", Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 570, p. 177-306, (1977))