Lifting of Nichols algebras of type \(B_2\). (With an appendix ``A generalization of the \(q\)-binomial theorem'' with I. Rutherford).
type: Item
instance of: scholarly article
title: Lifting of Nichols algebras of type \(B_2\). (With an appendix ``A generalization of the \(q\)-binomial theorem'' with I. Rutherford).
zbMATH Open document ID: 1054.16027
DOI: 10.1007/BF02784503
published in: Israel Journal of Mathematics
publication date: 2003-03-11T00:00:00Z
review text: Let \(\Gamma\) be a group, \(\mathbb{C}\Gamma\) its group algebra, \(V\) a Yetter-Drinfeld module over \(\Gamma\) and \({\mathfrak B}(V)\) its Nichols algebra. One of the main steps in the lifting method for the classification of complex pointed Hopf algebras is the determination of all Hopf algebras \(H\) such that \(\mathfrak H\) is of the form \(\mathbb{C}\Gamma\#{\mathfrak B}(V)\). See the exposition by \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [in New directions in Hopf algebras. Cambridge: Cambridge University Press. Math. Sci. Res. Inst. Publ. 43, 1-68 (2002; Zbl 1011.16025)]. In [loc. cit.], this determination is given when \(V\) is of Cartan type \(A_n\). The authors solve in the present paper the problem in the case when \(V\) is of Cartan type \(B_2\), and also some cases in type \(A_2\) left open in previous work by H.-J. Schneider and the reviewer. An interesting generalization of the quantum binomial formula -- needed for the proof of the main result -- is presented in an Appendix written in collaboration with I. Rutherford.
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 1874385
zbMATH Keywords: pointed Hopf algebras
zbMATH Keywords: Yetter-Drinfeld modules
zbMATH Keywords: Nichols algebras
zbMATH Keywords: quantum binomials
author: Margaret Beattie
author: Sorin Dascalescu
author: Serban Raianu
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
cites work: Q5301202
cites work: Q4944224
cites work: Hopf algebras of order \(p^2\) and braided Hopf algebras of order \(p\)
cites work: Lifting of quantum linear spaces and pointed Hopf algebras of order \(p^3\)
cites work: Finite quantum groups and Cartan matrices
cites work: Lifting of Nichols algebras of type \(A_2\) and pointed Hopf algebras of order \(p^4\)
cites work: On the number of types of finite dimensional Hopf algebras
cites work: Constructing pointed Hopf algebras by Ore extensions
cites work: Pointed Hopf algebras and Kaplansky's 10th conjecture
cites work: Pointed hopf algebras of dimension 32
cites work: Q4317887
cites work: Defending the negated Kaplansky conjecture
cites work: Q4274976
cites work: Finite subgroups of the quantum general linear group
cites work: Bialgebras of type one*
cites work: PBW-bases of quantum groups.
cites work: Hopf bigalois extensions
cites work: Q5590089
description: scientific article; zbMATH DE number 1874385