DOI QR코드

DOI QR Code

MONOIDAL FUNCTORS AND EXACT SEQUENCES OF GROUPS FOR HOPF QUASIGROUPS

  • Alvarez, Jose N. Alonso (Departamento de Matematicas Universidad de Vigo Campus Universitario Lagoas-Marcosende) ;
  • Vilaboa, Jose M. Fernandez (Departamento de Matematicas Universidad de Santiago de Compostela) ;
  • Rodriguez, Ramon Gonzalez (Departamento de Matematica Aplicada II Universidad de Vigo Campus Universitario Lagoas-Marcosende)
  • Received : 2020.02.10
  • Accepted : 2020.10.12
  • Published : 2021.03.01

Abstract

In this paper we introduce the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic(HMod) ≅ Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.

Keywords

Acknowledgement

The authors were supported by Ministerio de Economía, Industria y Competitividad of Spain. Agencia Estatal de Investigación. Unión Europea - Fondo Europeo de Desarrollo Regional. Grant MTM2016-79661-P: Homología, homotopía e invariantes categóricos en grupos y álgebras no asociativas.

References

  1. J. N. Alonso Alvarez, J. M. Fernandez Vilaboa, R. Gonzalez Rodriguez, and C. Soneira Calvo, Projections and Yetter-Drinfel'd modules over Hopf (co)quasigroups, J. Algebra 443 (2015), 153-199. https://doi.org/10.1016/j.jalgebra.2015.07.007
  2. J. N. Alonso Alvarez, J. M. Fernandez Vilaboa, M. P. Lopez Lopez, E. Villanueva Novoa, and R. Gonzalez Rodriguez, A Picard-Brauer five term exact sequence for braided categories, in Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), 11-41, Lecture Notes in Pure and Appl. Math., 197, Dekker, New York, 1998.
  3. J. N. Alonso Alvarez, R. Gonzalez Rodriguez, and J. M. Fernandez Vilaboa, The group of strong Galois objects associated to a cocommutative Hopf quasigroup, J. Korean Math. Soc. 54 (2017), no. 2, 517-543. https://doi.org/10.4134/JKMS.j160118
  4. J. M. Barja Perez, Morita theorems for triples in closed categories, Departamento de Algebra y Fundamentos, Universidad de Santiago de Compostela, Santiago, 1977.
  5. H. Bass, Algebraic K-Theory, W. A. Benjamin, Inc., New York, 1968.
  6. R. H. Bruck, Contributions to the theory of loops, Trans. Amer. Math. Soc. 60 (1946), 245-354. https://doi.org/10.2307/1990147
  7. T. Brzezinski, Hopf modules and the fundamental theorem for Hopf (co)quasigroups, Int. Electron. J. Algebra 8 (2010), 114-128.
  8. T. Brzezinski and Z. Jiao, Actions of Hopf quasigroups, Comm. Algebra 40 (2012), no. 2, 681-696. https://doi.org/10.1080/00927872.2010.535588
  9. S. Caenepeel, Computing the Brauer-Long group of a Hopf algebra. I. The cohomological theory, Israel J. Math. 72 (1990), no. 1-2, 38-83. https://doi.org/10.1007/BF02764611
  10. S. Caenepeel, Brauer groups, Hopf algebras and Galois theory, K-Monographs in Mathematics, 4, Kluwer Academic Publishers, Dordrecht, 1998.
  11. F. DeMeyer and E. Ingraham, Separable algebras over commutative rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, Berlin, 1971.
  12. J. M. Fernandez Vilaboa, E. Villanueva Novoa, and R. Gonzalez Rodriguez, Exact sequences for the Galois group, Comm. Algebra 24 (1996), no. 11, 3413-3435. https://doi.org/10.1080/00927879608825758
  13. C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995. https://doi.org/10.1007/978-1-4612-0783-2
  14. J. Klim, Integral theory for Hopf quasigroups, arXiv 1004.3929v2 (2010).
  15. J. Klim and S. Majid, Hopf quasigroups and the algebraic 7-sphere, J. Algebra 323 (2010), no. 11, 3067-3110. https://doi.org/10.1016/j.jalgebra.2010.03.011
  16. M. P. Lopez Lopez and E. Villanueva Novoa, The antipode and the (co)invariants of a finite Hopf (co)quasigroup, Appl. Categ. Structures 21 (2013), no. 3, 237-247. https://doi.org/10.1007/s10485-011-9260-5
  17. J. M. Perez-Izquierdo, Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math. 208 (2007), no. 2, 834-876. https://doi.org/10.1016/j.aim.2006.04.001
  18. J. M. Perez-Izquierdo and I. P. Shestakov, An envelope for Malcev algebras, J. Algebra 272 (2004), no. 1, 379-393. https://doi.org/10.1016/S0021-8693(03)00389-2