Lifting of Nichols algebras of type \(B_2\). (With an appendix “A generalization of the \(q\)-binomial theorem” with I. Rutherford). (English) Zbl 1054.16027
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 N. Andruskiewitsch and 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.
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.
Reviewer: Nicolás Andruskiewitsch (Cordoba)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
Citations:
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