An open problem in the theory of Hopf algebras is the classification of pointed Hopf algebras.
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
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 known as Nichols algebras. Nichols algebras are constructed from braided vector spaces. The braided vector spaces interesting for the classification mentioned are Yetter-Drinfeld modules 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
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
- Heckenberger, I. Classification of arithmetic root systems. Adv. Math. 220 (2009), no. 1, 59--124. MR2462836, link
- 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
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.
Small comment. The Weyl groupoid is an analogue of the usual Weyl group. It also works for Lie super algebras, see this MO Question.
Update. Another interesting open problem related to pointed Hopf algebra is a conjecture of Andruskiewitsch and Schneider related to generation in degree one. For some information and a categorical generalization see page 109 of:
- Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp. ISBN: 978-1-4704-2024-6 MR3242743
The conjecture is known to be true in several cases. For example, it is true for finite-dimensional pointed Hopf algebras with abelian coradical:
- Angiono, Iván. On Nichols algebras of diagonal type. J. Reine Angew. Math. 683 (2013), 189--251. MR3181554, link