# Classification of quasitriangular Hopf algebras

The classification of hopf algebras is a big and open problem, containing various subproblems (such as: the classification of groups, of Lie algebras, the study of special classes such as (co)commutative, (co)semisimple, pointed Hopf algebras etc).

One of the first results, known since the sixties, had to do with the classification of the cocommutative Hopf algebras:

Let $k$ an algebraically closed field, of characteristic zero, and let $H$ a cocommutative Hopf algebra (over $k$). If we denote by $G(H)$ the group of its grouplike elements and $P(H)$ the Lie algebra of its primitive elements, we have the following isomorphism of Hopf algebras $$H\cong U(P(H))\sharp kG(H)$$ where $\sharp$ stands for the smash product Hopf algebra. The above isomorphism is given explicitly by $$x_{i_{1}} \otimes x_{i_{2}} \otimes \ldots \otimes x_{i_{n}} \sharp g \mapsto x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}g$$ where, $i_{1} \leqslant \ldots \leqslant i_{n} \in I$, $n \in \mathbb{N}$, $I$ is a totally ordered set and: $\{x_{i}\}_{i \in I}$ are primitive elements of $Η$ forming a $k-$basis of $P(H)$.
(In the RHS of the last map, $x_{i_{1}}x_{i_{2}}\ldots x_{i_{n}}g$ stands for the product of the elements $x_{i_{1}} , x_{i_{2}} , \ldots , x_{i_{n}} , g$ inside $Η$).

The above result is frequently referred to, in the literature, as "the Cartier-Konstant-Milnor-Moore theorem". (although it seems that quite a lot of people have contributed to it).

Now, my question is: Given that the notion of quasitriangularity extends the notion of cocommutativity (in the sense that cocommutative hopf algebras are trivially quasitriangular through the $R$-matrix $R=1\otimes 1$ and thus quasitriangular are a class of Hopf algebras extending the class of cocommutative hopf algebras) are there similar results to the above, generalizing the Cartier-Konstant-Milnor-Moore theorem for quasitriangular Hopf algebras over an algebraically closed field of characteristic zero?

## 1 Answer

The best I know of are some classification results for triangular Hopf algebras, which would be a subcase. These are found in several papers by Etingof and/or Gelaki. See this paper and its references, for example. Theorem 2.2.2.4 therein is a result of Kostant that generalizes your quoted result, I'll note. I don't think even the triangular case has been completely determined, though I won't profess to be certain.

There are also some classification results on pointed quasitriangular Hopf algebras, such as this paper on minimal such ones generated by skew primitives by Masuoka.

Update (5/2/17): For some reason, an update to an old (1999) preprint of Etingof and Gelaki has been posted to the arxiv: The Classification of Triangular Semisimple and Cosemisimple Hopf Algebras Over an Algebraically Closed Field. Not sure if that was some automatic system recompiling of the article or legit update at this point. Either way, it's a paper worth looking at for the triangular case.