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

  • Iván AngionoEmail author
  • Agustín García Iglesias


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




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.


  1. 1.
    Andruskiewitsch, N., Angiono, I.: Finite dimensional Nichols algebras of diagonal type. Bull. Math. Sci. (to appear) arXiv:1707.08387
  2. 2.
    Andruskiewitsch, N., Angiono, I.: On Nichols algebras over basic Hopf algebras. arXiv:1802.00316
  3. 3.
    Andruskiewitsch, N., Schneider, H.-J.: Isomorphism classes and automorphisms of finite Hopf algebras of type \(A_n\). Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Biblioteca de la Revista Matematica Iberoamericana, pp. 201–226. Revista Matemática Iberoamericana, Madrid (2007)Google Scholar
  4. 4.
    Andruskiewitsch, N., García Iglesias, A.: Twisting Hopf algebras from cocycle deformations. Ann. Univ. Ferrara 63(2), 221–247 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups over abelian groups of prime exponent. Ann. Sci. Ec. Norm. Super. 35, 1–26 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171, 375–417 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Andruskiewitsch, N., Vay, C.: Finite dimensional Hopf algebras over the dual group algebra of the symmetric group in three letters. Commun. Algebra 39, 4507–4517 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Andruskiewitsch, N., Angiono, I., García Iglesias, A., Masuoka, A., Vay, C.: Lifting via cocycle deformation. J. Pure Appl. Algebra 218(4), 684–703 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Andruskiewitsch, N., Angiono, I., García Iglesias, A.: Liftings of Nichols algebras of diagonal type I. Cartan type A. Int. Math. Res. Not. IMRN 2017(9), 2793–2884 (2017)MathSciNetGoogle Scholar
  10. 10.
    Angiono, I.: On Nichols algebras of diagonal type. J. Reine Angew. Math. 683, 189–251 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Angiono, I.: Distinguished Pre-Nichols algebras. Transf. Groups 21, 1–33 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Angiono, I., García Iglesias, A.: Pointed Hopf algebras with standard braiding are generated in degreeone. Contemp. Math. 537, 57–70 (2011)CrossRefGoogle Scholar
  13. 13.
    Bergman, G.: The diamond lemma for ring theory. Adv. Math. 2(9), 178–218 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cohen, A.M., Gijsbers, D.A.H.: GBNP 0.9.5 (Non-commutative Gröbner bases).
  15. 15.
    Günther, R.: Crossed products for pointed Hopf algebras. Commun. Algebra 27, 4389–4410 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220, 59–124 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jury Giraldi, J.M., García Iglesias, A.: Liftings of Nichols algebras of diagonal type III. Cartan type \(G_2\). J. Algebra 478, 506–568 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Montgomery, S.: Hopf Algebras and Their Action on Rings. CBMS Lecture Notes, vol. 82. American Mathematical Society, Providence (1993)CrossRefGoogle Scholar
  19. 19.
    Schauenburg, P.: Hopf bi-Galois extensions. Commun. Algebra 24, 3797–3825 (1996)MathSciNetCrossRefGoogle Scholar
  20. 20.
    The GAP Group: GAP—Groups, Algorithms and Programming. Version 4.4.12 (2008).

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Authors and Affiliations

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

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