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On the classification of Galois objects over the quantum group of a nondegenerate bilinear form

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Abstract

We study Galois and bi-Galois objects over the quantum group of a nondegenerate bilinear form, including the quantum group \({\mathcal{O}_q}\) (SL(2)). We obtain the classification of these objects up to isomorphism and some partial results for their classification up to homotopy.

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References

  1. Bichon J. (2003) The representation category of the quantum group of a non-degenerate bilinear form. Comm. Algebra 31(10):4831–4851

    MathSciNet  Google Scholar 

  2. Bichon, J.: Galois and bi-Galois objects over monomial non-semisimple Hopf algebras. J. Algebra Appl. (to appear) (2006)

  3. Bichon J., Carnovale G. (2006) Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebra. J. Pure Appl. Algebra 204(3):627–665

    Article  MathSciNet  Google Scholar 

  4. Dubois-Violette M., Launer G. (1990) The quantum group of a nondegenerate bilinear form. Phys. Lett. B 245(2):175–177

    Article  MathSciNet  Google Scholar 

  5. Kassel C. (2004) Quantum principal bundles up to homotopy equivalence. In: Laudal O.A., Piene R. (eds) The Legacy of Niels Henrik Abel, The Abel Bicentennial Oslo, 2002. Springer, Berlin Heidelberg New York, pp. 737–748

    Google Scholar 

  6. Kassel C. (1995) Quantum Groups. Grad. Texts in Math., vol. 155. Springer, Berlin Heidelberg New York

    Google Scholar 

  7. Kassel C., Schneider H.-J. (2005) Homotopy theory of Hopf Galois extensions. Annales Inst. Fourier (Grenoble), 55:2521–2550

    MathSciNet  Google Scholar 

  8. Masuoka A. (1994) Cleft extensions for a Hopf algebra generated by a nearly primitive element. Comm. Algebra 22(11):4537–5459

    MathSciNet  Google Scholar 

  9. Masuoka, A.: Cocycle deformations and Galois objects for some Cosemisimple Hopf Algebras of finite dimension. In: New trends in Hopf algebra theory (La Falda, 1999), pp. 195–214. Contemp. Math. vol. 267. Amer. Math. Soc., Providence, RI (2000)

  10. Montgomery, S.: Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, vol. 82. Amer. Math. Soc., Providence, RI (1993)

  11. Ostrik: Module categories over representations of SL q (2) and graphs. Preprint arXiv: math.QA/0509530

  12. Panaite F., Van Oystaeyen F. (2000) Clifford type algebras as cleft extensions for some pointed Hopf algebras. Comm. Algebra 28(2):585–600

    MathSciNet  Google Scholar 

  13. Riehm C. (1974) The equivalence of bilinear forms. J. Algebra 31, 45–66

    Article  MathSciNet  Google Scholar 

  14. Schauenburg P. (1996) Hopf bi-Galois extensions. Comm. Algebra 24(12):3797–3825

    MathSciNet  Google Scholar 

  15. Schauenburg P. (2000) BiGalois objects over the Taft algebras. Israel J. Math. 115, 101–123

    MathSciNet  Google Scholar 

  16. Schauenburg, P.: Hopf Galois and bi-Galois extensions. In: Galois theory, Hopf algebras and semiabelian categories, pp. 469–515. Fields Inst. Commun., vol. 43. Amer. Math. Soc., Providence, RI (2004)

  17. Schneider H.-J. (1990) Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72(1–2):167–195

    MathSciNet  Google Scholar 

  18. Ulbrich K.-H. (1987) Galois extensions as functors of comodules. Manuscripta Math. 59(4):391–397

    Article  MathSciNet  Google Scholar 

  19. Ulbrich K.-H. (1989) Fibre functors of finite-dimensional comodules. Manuscripta Math. 65(1):39–46

    Article  MathSciNet  Google Scholar 

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  1. Institut de Recherche Mathématique Avancée, C.N.R.S., Université Louis Pasteur, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Thomas Aubriot

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  1. Thomas Aubriot
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Correspondence to Thomas Aubriot.

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Aubriot, T. On the classification of Galois objects over the quantum group of a nondegenerate bilinear form. manuscripta math. 122, 119–135 (2007). https://doi.org/10.1007/s00229-006-0054-2

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  • DOI: https://doi.org/10.1007/s00229-006-0054-2

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