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 \({\text {gr}}\mathfrak {u}(\varvec{\lambda })\simeq \mathfrak {B}(V)\# H\).
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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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The work was partially supported by CONICET, FONCyT-ANPCyT, Secyt (UNC), the MathAmSud project GR2HOPF.
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Angiono, I., García Iglesias, A. Liftings of Nichols algebras of diagonal type II: all liftings are cocycle deformations. Sel. Math. New Ser. 25, 5 (2019). https://doi.org/10.1007/s00029-019-0452-4
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DOI: https://doi.org/10.1007/s00029-019-0452-4