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Problem 1. Classify finite-dimensional Nichols algebras.

 

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

Problem 1. Classify finite-dimensional Nichols algebras.

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

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

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.

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

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