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POINTED HOPF ACTIONS ON CENTRAL SIMPLE DIVISION ALGEBRAS

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Abstract

We examine actions of finite-dimensional pointed Hopf algebras on central simple division algebras in characteristic 0. (By a Hopf action we mean a Hopf module algebra structure.) In all examples considered, we show that the given Hopf algebra does admit a faithful action on a central simple division algebra, and we construct such a division algebra. This is in contrast to earlier work of Etingof and Walton, in which it was shown that most pointed Hopf algebras do not admit faithful actions on fields. We consider all bosonizations of Nichols algebras of finite Cartan type, small quantum groups, generalized Taft algebras with non-nilpotent skew primitive generators, and an example of non-Cartan type.

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References

  1. Andruskiewitsch, N., Angiono, I.: On finite dimensional Nichols algebras of diagonal type. Bull. Math. Sci. 7(3), 353–573 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Andruskiewitsch, N., Schneider, H.-J.: Finite quantum groups over abelian groups of prime exponent. Ann. Sci. Éc. Norm. Supfier. 35(1), 1–26 (2002)

    Article  MathSciNet  Google Scholar 

  4. Andruskiewitsch, N., Schneider, H.-J.: On the classification of finite-dimensional pointed Hopf algebras. Ann. of Math. 171(1), 375–417 (2010)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Artamonov, V.A.: Actions of pointed Hopf algebras on quantum torus. Ann. Univ. Ferrara. 51(1), 29–60 (2005)

    Article  MathSciNet  Google Scholar 

  8. Brown, K., Goodearl, K.R.: Lectures on Algebraic Quantum Groups, Advanced Courses in Mathematics. Birkhäuser Verlag, Basel (2012)

    Google Scholar 

  9. Cohen, M., Fischman, D., Montgomery, S.: Hopf Galois extensions, smash products, and Morita equivalence. J. Algebra. 133(2), 351–372 (1990)

    Article  MathSciNet  Google Scholar 

  10. Cuadra, J., Etingof, P.: Finite dimensional Hopf actions on central division algebras. Int. Math. Res. Not. IMRN. 2017(5), 1562–1577 (2017)

    MathSciNet  MATH  Google Scholar 

  11. C. De Concini, V. G. Kac, Representations of quantum groups at roots of 1, in: Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris, 1989), Progr. Math., Vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 471–506.

  12. Etingof, P.: Galois bimodules and integrality of PI comodule algebras over invariants. J. Noncommut. Geom. 9(2), 567–602 (2015)

    Article  MathSciNet  Google Scholar 

  13. Etingof, P., Walton, C.: Pointed Hopf actions on fields, I. Transform. Groups. 20(4), 985–1013 (2015)

    Article  MathSciNet  Google Scholar 

  14. Etingof, P., Walton, C.: Finite dimensional Hopf actions on algebraic quantizations. Algebra Number Theory. 10(10), 2287–2310 (2016)

    Article  MathSciNet  Google Scholar 

  15. Etingof, P., Walton, C.: Pointed Hopf actions on fields, II. J. Algebra. 460, 253–283 (2016)

    Article  MathSciNet  Google Scholar 

  16. Graña, M.: On Nichols algebras of low dimension. Contemp. Math. 267, 111–134 (2000)

    Article  MathSciNet  Google Scholar 

  17. L. Grunenfelder, M. Mastnak, Pointed Hopf algebras as cocycle deformations, arXiv: 1010.4976 (2010).

  18. Heckenberger, I.: The Weyl groupoid of a Nichols algebra of diagonal type. Invent. Math. 164(1), 175–188 (2006)

    Article  MathSciNet  Google Scholar 

  19. Heckenberger, I.: Classification of arithmetic root systems. Adv. Math. 220(1), 59–124 (2009)

    Article  MathSciNet  Google Scholar 

  20. Heckenberger, I.: Lusztig isomorphisms for Drinfel’d doubles of bosonizations of Nichols algebras of diagonal type. J. Algebra. 323(8), 2130–2182 (2010)

    Article  MathSciNet  Google Scholar 

  21. Heyneman, R., Radford, D.: Reexivity and coalgebras of finite type. J. Algebra. 28(2), 215–246 (1974)

    Article  MathSciNet  Google Scholar 

  22. Lusztig, G.: Quantum groups at roots of 1. Geom. Dedicata. 35(1), 89–113 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Lusztig, G.: Introduction to Quantum Groups, Progress in Mathematics, vol. 110. Birkhäuser Boston, Boston, MA (2010)

    Book  Google Scholar 

  24. Mastnak, M., Pevtsova, J., Schauenburg, P., Witherspoon, S.: Cohomology of finite-dimensional pointed hopf algebras. Proc. Lond. Math. Soc. 100(2), 377–404 (2010)

    Article  MathSciNet  Google Scholar 

  25. Masuoka, A.: Abelian and non-abelian second cohomologies of quantized enveloping algebras. J. Algebra. 320(1), 1–47 (2008)

    Article  MathSciNet  Google Scholar 

  26. Masuoka, A., Wigner, D.: Faithful flatness of Hopf algebras. J. Algebra. 170(1), 156–164 (1994)

    Article  MathSciNet  Google Scholar 

  27. S. Montgomery, Hopf Algebras and Their Actions on Rings, CBMS Regional Conference Series in Mathematics, Vol. 82, American Mathematical Society, Providence, RI, 1993.

  28. Negron, C.: Braided Hochschild cohomology and Hopf actions. J. Noncommut. Geom. 13(1), 1–33 (2019)

    Article  MathSciNet  Google Scholar 

  29. Nichols, W.D.: Bialgebras of type one. Comm. Algebra. 6(15), 1521–1552 (1978)

    Article  MathSciNet  Google Scholar 

  30. Skryabin, S., Van Oystaeyen, F.: The Goldie theorem for H-semiprime algebras. J. Algebra. 05(1), 292–320 (2006)

    Article  MathSciNet  Google Scholar 

  31. Taft, E.J., Wilson, R.L.: On antipodes in pointed Hopf algebras. J. Algebra. 29(1), 27–32 (1974)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    P. ETINGOF

  2. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    C. NEGRON

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  1. P. ETINGOF
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  2. C. NEGRON
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Correspondence to C. NEGRON.

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P. Etingof is supported by NSF grant DMS-1502244.

C. Negron is supported by NSF Postdoctoral Research Fellowship DMS-1503147.

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ETINGOF, P., NEGRON, C. POINTED HOPF ACTIONS ON CENTRAL SIMPLE DIVISION ALGEBRAS. Transformation Groups 27, 471–495 (2022). https://doi.org/10.1007/s00031-021-09690-9

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