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, 25:5 | Cite as

Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations

  • Iván AngionoEmail author
  • Agustín García Iglesias
Article
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Abstract

We classify finite-dimensional pointed Hopf algebras with abelian coradical, up to isomorphism, and show that they are cocycle deformations of the associated graded Hopf algebra. More generally, for any braided vector space of diagonal type V with a principal realization in the category of Yetter–Drinfeld modules of a cosemisimple Hopf algebra H and such that the Nichols algebra \(\mathfrak {B}(V)\) is finite-dimensional, thus presented by a finite set \({{\mathcal {G}}}\) of relations, we define a family of Hopf algebras \(\mathfrak {u}(\varvec{\lambda })\), \(\varvec{\lambda }\in \Bbbk ^{{{\mathcal {G}}}}\), which are cocycle deformations of \(\mathfrak {B}(V)\# H\) and such that \({\text {gr}}\mathfrak {u}(\varvec{\lambda })\simeq \mathfrak {B}(V)\# H\).

Mathematics Subject Classification

16T05 

Notes

Acknowledgements

We thank Nicolás Andruskiewitsch for his constant support and council. We also thank Cristian Vay for pointing to us a mistake in a previous version of this article. We thank the referee for his/her comments, that we believe have helped to improve the presentation of the article, as well as the scope of potential readers.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.FaMAF-CIEM (CONICET)Universidad Nacional de CórdobaCórdobaArgentina

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