On Nichols algebras of low dimension (Q2708911)

The most efficient method for classifying finite dimensional pointed Hopf algebras over an algebraically closed field \(k\) of characteristic 0 has proved to be the ``Lifting Method'', invented by \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [J. Algebra 209, No. 2, 659-691 (1998; Zbl 0919.16027)]. The first step of this method is the classification of Nichols algebras of certain dimensions over the group algebra of a finite group. Such a Nichols algebra is a positively graded Hopf algebra \(R\) in the category of Yetter-Drinfeld modules over a finite dimensional group algebra, such that \(R_0=k\), \(R_1\) is the space of primitive elements of \(R\), and \(R\) is generated as an algebra by \(R_1\). The author gives a generalization of the quantum Serre relations and proposes a generalization of the Frobenius-Lusztig kernels, and he uses these to classify Nichols algebras with dimension \(<32\) or with dimension \(p^3\), \(p\) a prime number. This allows the classification of pointed Hopf algebras with coradical of index either \(<32\) or \(p^3\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].