An open problem in the theory of Hopf algebras is the classification of [pointed Hopf algebras][1]. 

One method to classify finite-dimensional pointed Hopf
algebras is the Lifting Method of Andruskiewitsch and Schneider. The method was
proved to be successful in the case of abelian coradical, see for example

* Andruskiewitsch, Nicolás; Schneider, Hans-Jürgen. On the classification of finite-dimensional pointed Hopf algebras. Ann. of Math. (2) 171 (2010), no. 1, 375--417. MR2630042, [doi][2] 

The problem in the case where the coradical is a non-abelian group is still open. 

I will be more precise. The heart of the method is the understanding of the structure of certain
finite-dimensional [braided Hopf algebras][3]
known as [Nichols algebras][4]. Nichols algebras are constructed from [braided vector spaces][5]. The braided vector spaces
interesting for the classification mentioned are [Yetter-Drinfeld modules][6] over groups. 

In the survey

 - Andruskiewitsch, Nicolás. About finite dimensional Hopf algebras. Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), 1--57, Contemp. Math., 294, Amer. Math. Soc., Providence, RI, 2002. MR1907185, [link][7]

one finds the following problems:

> **Problem 1.** Classify finite-dimensional Nichols algebras.

> **Problem 2.** Obtain a "nice" presentation by generators and relations of
> finite-dimensional Nichols algebras.

These problems are in general open; they were solved in the case of braided vector spaces of diagonal
type, i.e. Yetter-Drinfeld modules over abelian groups.  The first one was
solved by Heckenberger; the second one, by Angiono.  Both solutions deeply use the so-called 
Weyl groupoid. References: 

* Heckenberger, I. The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164 (2006), no. 1, 175--188. MR2207786, [link][8]
* Heckenberger, I. Classification of arithmetic root systems. Adv. Math. 220 (2009), no. 1, 59--124. MR2462836, [link][9]
* Angiono, Iván Ezequiel. A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems. J. Eur. Math. Soc. (JEMS) 17 (2015), no. 10, 2643--2671. MR3420518, [link][10]

So I would add the following as an problem in the theory of Hopf algebras:

> Classify finite-dimensional Nichols algebras over non-abelian groups.

Partial results are known. However, several questions are still open. An interesting particular case is related to symmetric groups. This particular problem is connected to
some quadratic algebras known as [Fomin-Kirillov algebras][11].

**Small comment.** The Weyl groupoid is an analogue of the usual Weyl group. It also works for Lie super algebras, see [this MO Question][12].


  [1]: http://mathoverflow.net/questions/86627/what-is-a-pointed-hopf-algebra
  [2]: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.4007/annals.2010.171.375
  [3]: https://en.wikipedia.org/wiki/Braided_Hopf_algebra
  [4]: https://en.wikipedia.org/wiki/Nichols_algebra
  [5]: https://en.wikipedia.org/wiki/Braided_vector_space
  [6]: https://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld_category
  [7]: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1090/conm/294/04969
  [8]: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1007/s00222-005-0474-8
  [9]: http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.1016/j.aim.2008.08.005
  [10]: http://www.ams.org/leavingmsn?url=http://www.ems-ph.org/journals/all_issues.php?issn=1435-9855
  [11]: http://mathoverflow.net/questions/152774/fomin-kirillov-algebras-and-schubert-calculus
  [12]: http://mathoverflow.net/questions/26304/is-there-a-definition-of-analogue-weyl-group-for-lie-super-algebra