I assume that the classification of (certain families of) Hopf algebras is still an open problem, am I right?


My question is the following: What is the current state of the art? What is known about the classification of certain families of Hopf algebras? (To be more precise, say for example that I am interested in finite-dimensional or combinatorial Hopf algebras.)



Classification of Hopf algebras (state of the art)
The general problem about the classification of finite-dimensional Hopf
algebras (over $\mathbb{C}$) is widely open. I mention some general results.


Here


Zhu, Yongchang. Hopf algebras of prime dimension. Internat. Math. Res. Notices 1994, no. 1, 53--59. MR1255253 (94j:16072), [link](https://doi.org/10.1155/S1073792894000073)


it is proved that a Hopf algebra of prime dimension is isomorphic to a group
algebra. Here


Ng, Siu-Hung. Non-semisimple Hopf algebras of dimension $p^2$. J. Algebra 255 (2002), no. 1, 182--197. MR1935042 (2003h:16067), [link](https://doi.org/10.1016/S0021-8693(02)00139-4)


it is proved that the only Hopf algebras of dimension $p^2$ are the group
algebras and the Taft algebras. Hopf algebras of dimension $2p^2$ were also
classified by Hilgemann and Ng, see the following paper:


Hilgemann, Michael; Ng, Siu-Hung. Hopf algebras of dimension $2p^2$. J.
Lond. Math. Soc. (2) 80 (2009), no. 2, 295--310. MR2545253 (2010h:16080), [link](https://doi.org/10.1112/jlms/jdp026)


Of course, these are not the only general results. For a good accound related
to the classification of Hopf algebras of a given dimension you may want to check
the following papers:


Beattie, Margaret. A survey of Hopf algebras of low dimension. Acta Appl. Math.
108 (2009), no. 1, 19--31. MR2540955 (2010i:16054), [link](https://doi.org/10.1007/s10440-008-9367-3)


M. Beattie and G. A. García. Classifying Hopf algebras of a given dimension.
Preprint: [arXiv:1206.6529](https://arxiv.org/abs/1206.6529)


It is important to mention that it is very interesting to study the classification of certain families of finite-dimensional Hopf algebras. If this is your interest, maybe google can help.


