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Examples of liftings of Nichols algebras over racks. (English) Zbl 1032.16029
One of the crucial points in the Lifting Method for classification of pointed Hopf algebras is the determination of the Nichols algebras of suitable braided vector spaces [N. Andruskiewitsch, H.-J. Schneider, in New directions in Hopf algebras, Math. Sci. Res. Inst. Publ. 43, 1-68 (2002; Zbl 1011.16025)]. In the case of pointed Hopf algebras with non-Abelian group of group-likes, only a few examples of Nichols algebras are known to be finite dimensional; some of them appear in the work of the authors [Adv. Math. 178, No. 2, 177-243 (2003; see the preceding review Zbl 1032.16028)], connecting the problem to rack cohomology, and in the paper [Contemp. Math. 267, 215-236 (2000; Zbl 1093.16504)] by A. Milinski and H.-J. Schneider. In the paper under review the authors first list these few known examples. The paper also contains some new contributions which consist in performing the remaining steps in the Lifting Method for some of these examples: it is shown that generation by group-likes and skew-primitives holds for pointed Hopf algebras with infinitesimal braiding of these types; all the liftings in the cases of $$S_3$$ and $$S_4$$ are also computed.

MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations
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