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On Nichols algebras over basic Hopf algebras

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Abstract

This is a contribution to the classification of finite-dimensional Hopf algebras over an algebraically closed field \(\mathbb {k}\) of characteristic 0. Concretely, we show that a finite-dimensional Hopf algebra whose Hopf coradical is basic is a lifting of a Nichols algebra of a semisimple Yetter–Drinfeld module and we explain how to classify Nichols algebras of this kind. We provide along the way new examples of Nichols algebras and Hopf algebras with finite Gelfand–Kirillov dimension.

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References

  1. Andruskiewitsch, N.: Some Remarks on Nichols Algebras. In Bergen, Catoiu and Chin (eds) Hopf Algebras, pp. 25–45. M. Dekker, New York (2004)

    Google Scholar 

  2. Andruskiewitsch, N.: On finite-dimensional Hopf algebras. In: Proceedings of the ICM Seoul 2014 Vol. II, 117–141 (2014)

  3. Andruskiewitsch, N.: An introduction to Nichols Algebras. In: Cardona, A., Morales, P., Ocampo, H., Paycha, S., Reyes, A. (eds.) Quantization, Geometry and Noncommutative Structures in Mathematics and Physics, pp. 135–195. Springer, New York (2017)

    Google Scholar 

  4. Andruskiewitsch, N., Angiono, I.: On Nichols algebras with generic braiding. In: Brzezinski, T., Gomez Pardo, J.L., Shestakov, I., Smith, P.F. (eds.) Modules and Comodules, Trends in Mathematics, pp. 47–64. Birkhauser, Basel (2008)

    Google Scholar 

  5. Andruskiewitsch, N., Angiono, I.: On finite dimensional Nichols algebras of diagonal type. Bull. Math. Sci. 7, 353–573 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Andruskiewitsch, N., Angiono, I., Heckenberger, I.: On finite GK-dimensional Nichols algebras over abelian groups. Mem. Amer. Math. Soc. (to appear)

  7. Andruskiewitsch, N., Angiono, I., Heckenberger, I.: On Nichols algebras of infinite rank with finite Gelfand–Kirillov dimension. arXiv:1805.12000

  8. Andruskiewitsch, N., Cuadra, J.: On the structure of (co-Frobenius) Hopf algebras. J. Noncommut. Geom. 7, 83–104 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Andruskiewitsch, N., Heckenberger, I., Schneider, H.-J.: The Nichols algebra of a semisimple Yetter–Drinfeld module. Am. J. Math. 132, 1493–1547 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Andruskiewitsch, N., Schneider, H.-J.: Lifting of quantum linear spaces and pointed Hopf algebras of order \( p^3\). J. Algebra 209, 658–691 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups and Cartan matrices. Adv. Math. 154, 1–45 (2000)

    MathSciNet  MATH  Google Scholar 

  12. Andruskiewitsch, N., Schneider, H.J.: Pointed Hopf algebras. Recent Developments in Hopf Algebras Theory, vol. 43, pp. 1–68. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  13. Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. Math. 171, 375–417 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Angiono, I.: A presentation by generators and relations of Nichols algebras of diagonal type and convex orders on root systems. J. Eur. Math. Soc. 17, 2643–2671 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Angiono, I.: On Nichols algebras of diagonal type. J. Reine Angew. Math. 683, 189–251 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Angiono, I.: Distinguished pre-Nichols algebras. Transf. Groups 21, 1–33 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Angiono, I., García Iglesias, A.: Liftings of Nichols algebras of diagonal type II. All liftings are cocycle deformations. arXiv:1605.03113

  18. Angiono, I., García Iglesias, A.: Pointed Hopf algebras: a guided tour to the liftings. arXiv:1807.07154

  19. Artin, M., Schelter, W.F.: Graded algebras of global dimension 3. Adv. Math. 66, 171–216 (1987)

    MathSciNet  MATH  Google Scholar 

  20. Artin, M., Schelter, W.F., Tate, J.: Quantum deformations of \(GL_n\). Commun. Pure Appl. Math. 44, 879–895 (1991)

    MATH  Google Scholar 

  21. Artin, M., Small, L.W., Zhang, J.J.: Generic flatness for strongly Noetherian algebras. J. Algebra 221, 579–610 (1999)

    MathSciNet  MATH  Google Scholar 

  22. Cuntz, M., Lentner, S.: A simplicial complex of Nichols algebras. Math. Z. 285, 647–683 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Doi, Y., Takeuchi, M.: Multiplication alteration by two-cocycles—the quantum version. Commun. Algebra 22, 5715–5732 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Drinfeld, V.G.: Hopf algebras and the quantum Yang–Baxter equation. Sov. Math. Dokl. 32, 256–258 (1985)

    Google Scholar 

  25. Elle, S.: Classification of relation types of Ore extensions of dimension 5. Commun. Algebra 45, 1323–1346 (2017)

    MathSciNet  MATH  Google Scholar 

  26. García, G.A., Giraldi, J.M.J.: On Hopf Algebras over quantum subgroups. J. Pure Appl. Algebra 223, 738–768 (2019)

    MathSciNet  MATH  Google Scholar 

  27. García, G.A., Mastnak, M.: Deformation by cocycles of pointed Hopf algebras over non-abelian groups. Math. Res. Lett. 22, 59–92 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Graña, M.: A freeness theorem for Nichols algebras. J. Algebra 231, 235–257 (2000)

    MathSciNet  MATH  Google Scholar 

  29. Heckenberger, I.: The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164, 175–188 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220, 59–124 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Heckenberger, I., Schneider, H.-J.: Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid. Isr. J. Math. 197, 139–187 (2013)

    MathSciNet  MATH  Google Scholar 

  32. Heckenberger, I., Schneider, H.-J.: Yetter–Drinfeld modules over bosonizations of dually paired Hopf algebras. Adv. Math. 244, 54–394 (2013)

    MathSciNet  MATH  Google Scholar 

  33. Heckenberger, I., Vendramin, L.: A classification of Nichols algebras of semi-simple Yetter–Drinfeld modules over non-abelian groups. J. Eur. Math. Soc. 19, 299–356 (2017)

    MathSciNet  MATH  Google Scholar 

  34. Heckenberger, I., Yamane, H.: Drinfel’d doubles and Shapovalov determinants. Rev. Un. Mat. Argent. 51, 107–146 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Heckenberger, I., Yamane, H.: A generalization of Coxeter groups, root systems, and Matsumoto’s theorem. Math. Z. 259, 255–276 (2008)

    MathSciNet  MATH  Google Scholar 

  36. Hu, N., Xiong, R.: On families of Hopf algebras without the dual Chevalley property. Rev. Un. Mat. Argent. 59, 443–469 (2018)

    MathSciNet  MATH  Google Scholar 

  37. Lusztig, G.: Introduction to quantum groups. Birkhäuser, Basel (1993)

    MATH  Google Scholar 

  38. Li, J., Wang, X.: Some five-dimensional Artin–Schelter regular algebras obtained by deforming a Lie algebra. J. Algebra Appl. 15(04), 1650060 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Majid, S.: Doubles of quasitriangular Hopf algebras. Commun. Algebra 19, 3061–3073 (1991)

    MathSciNet  MATH  Google Scholar 

  40. Montgomery, S.: Hopf Algebras and their Actions on Rings, CMBS 82. American Mathematical Society, Providence (1993)

    Google Scholar 

  41. Nevins, T.A., Stafford, J.T.: Sklyanin algebras and Hilbert schemes of points. Adv. Math. 210, 405–478 (2007)

    MathSciNet  MATH  Google Scholar 

  42. Pogorelsky, B., Vay, C.: Verma and simple modules for quantum groups at non-abelian groups. Adv. Math. 301, 423–457 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Radford, D.E., Schneider, H.-J.: On the simple representations of generalized quantum groups and quantum doubles. J. Algebra 319, 3689–3731 (2008)

    MathSciNet  MATH  Google Scholar 

  44. Rosso, M.: Quantum groups and quantum shuffles. Invent. Math. 133, 399–416 (1998)

    MathSciNet  MATH  Google Scholar 

  45. Schauenburg, P.: Hopf bi-Galois extensions. Commun. Algebra 24, 3797–3825 (1996)

    MathSciNet  MATH  Google Scholar 

  46. Ufer, S.: PBW bases for a class of braided Hopf algebras. J. Algebra 280, 84–119 (2004)

    MathSciNet  MATH  Google Scholar 

  47. Ufer, S.: Triangular braidings and pointed Hopf algebras. J. Pure Appl. Algebra 210, 307–320 (2007)

    MathSciNet  MATH  Google Scholar 

  48. Ufer, S.: Braided Hopf algebras of triangular type. PhD thesis (2004). https://edoc.ub.uni-muenchen.de/2477/1/ufer_stefan.pdf

  49. Wang, Q., Wu, Q.S.: A class of AS-regular algebras of dimension five. J. Algebra 362, 117–144 (2012)

    MathSciNet  MATH  Google Scholar 

  50. Xiong, R.: On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property. Commun. Algebra 47, 1516–1540 (2019)

    MathSciNet  MATH  Google Scholar 

  51. Xiong, R.: Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra. arXiv:1801.06205

  52. Xiong, R.: On Hopf algebras over basic Hopf algebras of dimension 24. arXiv:1809.03938

  53. Zhang, J.J., Zhang, J.: Double extension regular algebras of type. J. Algebra 322, 373–409 (2009)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This paper grew from conversations following a talk by Oscar Márquez on joint work in progress with Dirceu Bagio and Gastón A. García at the Colloquium Quantum 17 hosted by the University of Talca (Chile). We thank them for sharing their results as well as María Ronco and María Inés Icaza for hospitality. We also thank Hiroyuki Yamane for pointing out to us the reference [48]. We are grateful to C. D. Ward and H. West (University of Miskatonic, Arkham) for pointing out to us a mistake in the proof of Lemma 3.6. The main results of this paper were communicated at the XXII Coloquio Latinoamericano de Álgebra (Quito, August 2017); the Reunión Anual de la Unión Matemática Argentina (Buenos Aires, December 2017); the Workshop Métodos Categóricos en Álgebras de Hopf (Maldonado, December 2017); the Workshop Tensor categories, Hopf algebras and quantum groups (Marburg, January 2018).

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  1. Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM-CONICET, Medina Allende s/n, Ciudad Universitaria, 5000, Córdoba, Argentina

    Nicolás Andruskiewitsch & Iván Angiono

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  1. Nicolás Andruskiewitsch
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Correspondence to Iván Angiono.

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The work of N. A. and I. A. was partially supported by CONICET, Secyt (UNC), the MathAmSud project GR2HOPF.

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Andruskiewitsch, N., Angiono, I. On Nichols algebras over basic Hopf algebras. Math. Z. 296, 1429–1469 (2020). https://doi.org/10.1007/s00209-020-02493-w

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