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THE GROUP OF BI-GALOIS OBJECTS OVER THE COORDINATE ALGEBRA OF THE FROBENIUS–LUSZTIG KERNEL OF SL(2)

Part of: Categories with structure Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  21 July 2015

JULIEN BICHON
JULIEN BICHON*
Affiliation:
Laboratoire de Mathématiques, Université Blaise Pascal, Complexe universitaire des Cézeaux, 63171 Aubière Cedex, France e-mail: Julien.Bichon@math.univ-bpclermont.fr
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Abstract

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We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over uq(sl(n))*, the coordinate algebra of the Frobenius–Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.

MSC classification

Primary: 16T05: Hopf algebras and their applications
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 727 - 738
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

1. Andruskiewitsch, N. and Devoto, J., Extensions of Hopf algebras, St. Petersburg Math. J. 7 (1) (1996), 17–52.Google Scholar
2. Arkhipov, S. and Gaitsgory, D., Another realization of the category of modules over the small quantum group, Adv. Math. 173 (1) (2003), 114–143.CrossRefGoogle Scholar
3. Aubriot, T., On the classification of Galois objects over the quantum group of a nondegenerate bilinear form, Manuscr. Math. 122 (1) (2007), 119–135.Google Scholar
4. Bichon, J., The representation category of the quantum group of a non–degenerate bilinear form, Comm. Algebra 31 (10) (2003), 4831–4851.Google Scholar
5. Bichon, J., Galois and bigalois objects over monomial non-semisimple Hopf algebras, J. Algebra Appl. 5 (5) (2006), 653–680.CrossRefGoogle Scholar
6. Bichon, J. and Carnovale, G., Lazy cohomology: An analogue of the Schur multiplier for arbitrary Hopf algebras, J. Pure Appl. Algebra 204 (3) (2006), 627–665.CrossRefGoogle Scholar
7. Bontea, C. G. and Nikshych, D., On the Brauer-Picard group of a finite symmetric tensor category, arXiv:1408.6445.Google Scholar
8. Caenepeel, S., Van Oystaeyen, F. and Zhang, Y. H., The Brauer group of Yetter-Drinfeld module algebras, Trans. Amer. Math. Soc. 349 (9) (1997), 3737–3771.Google Scholar
9. Carnovale, G. and Cuadra, J., On the subgroup structure of the full Brauer group of Sweedler Hopf algebra, Isr. J. Math. 183 (2011), 61–92.Google Scholar
10. Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups, Advanced Courses in Mathematics CRM Barcelona (Birkhäuser Verlag, Basel, 2002).Google Scholar
11. Davydov, A. and Nikshych, D., The Picard crossed module of a braided tensor category, Algebra Number Theory 7 (6) (2013), 1365–1403.CrossRefGoogle Scholar
12. Doi, Y. and Takeuchi, M., Hopf-Galois extensions of algebras, the Miyashita–Ulbrich action, and Azumaya algebras, J. Algebra 121 (2) (1989), 488–516.CrossRefGoogle Scholar
13. Etingof, P., Nikshych, D. and Ostrik, V., Fusion categories and homotopy theory, Quantum Topol. 1 (3) (2010), 209–273.Google Scholar
14. Günther, R., Crossed products for pointed Hopf algebras, Comm. Algebra 27 (9) (1999), 4389–4410.Google Scholar
15. Kassel, C., Quantum groups, GTM, vol. 155 (Springer, New York, 1995).CrossRefGoogle Scholar
16. Kreimer, H. F. and Cook, P. M. II, Galois theories and normal bases, J. Algebra 43 (1) (1976), 115–121.Google Scholar
17. Masuoka, A., Defending the negated Kaplansky conjecture, Proc. Amer. Math. Soc. 129 (11) (2001), 3185–3192.Google Scholar
18. Mejía, A. Castaño and M. Mombelli, Crossed extensions of the corepresentation category of finite supergroup algebras, arXiv:1405.0979.Google Scholar
19. Mombelli, M., The Brauer–Picard group of the representation category of finite supergroup algebras, Rev. Un. Mat. Argentina 55 (1) (2014), 83–117.Google Scholar
20. Montgomery, S., Hopf algebras and their actions on rings, (American Mathematical Society, Providence, 1993).Google Scholar
21. Neshveyev, S. and Tuset, L., Autoequivalences of the tensor category of $U_q \mathfrak g$ -modules, Int. Math. Res. Not. IMRN (15) (2012), 3498–3508.Google Scholar
22. Parshall, B. and Wang, J. P., Quantum linear groups, Mem. Amer. Math. Soc. 89 (439) (1991).Google Scholar
23. Schauenburg, P., Hopf bigalois extensions, Comm. Algebra 24 (12) (1996), 3797–3825.CrossRefGoogle Scholar
24. Schauenburg, P., Galois objects over generalized Drinfeld doubles, with an application to uq(sl2) , J. Algebra 217 (2) (1999), 584–598.Google Scholar
25. Schauenburg, P., BiGalois objects over the Taft algebras, Israel J. Math. 115 (2000), 101–123.Google Scholar
26. Schneider, H.-J., Principal homogeneous spaces for arbitrary Hopf algebras, Isr. J. Math. 72 (1–2) (1990), 167–195.Google Scholar
27. Schneider, H. J., Normal basis and transitivity of crossed products for Hopf algebras, J. Algebra 152 (2) (1992), 289–312.CrossRefGoogle Scholar
28. Takeuchi, M., Some topics on GL q (n), J. Algebra 147 (2) (1992), 379–410.Google Scholar
29. Van Oystaeyen, F. and Zhang, Y. H., The Brauer group of a braided monoidal category, J. Algebra 202 (1) (1998), 96–128.CrossRefGoogle Scholar