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H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

Year 2017, , 23 - 38, 17.01.2017
https://doi.org/10.24330/ieja.295749

Abstract

 Let H be a weak Hopf algebra and let A be an H-comodule algebra
with subalgebra of coinvariants AH. In this paper we introduce the
notion of H-Galois extension with normal basis and we prove that AH ,→ A
is an H-Galois extension with normal basis if and only if AH ,→ A is an
H-cleft extension which admits a convolution invertible total integral. As a
consequence, if H is cocommutative and A commutative, we obtain a bijective
correspondence between the second cohomology group H2
ϕAH
(H, AH) and the
set of isomorphism classes of H-Galois extensions with normal basis whose left
action over AH is ϕAH . 

References

  • [1] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and ´
  • A. B. Rodr´ıguez Raposo, Weak C-cleft extensions, weak entwining structures
  • and weak Hopf algebras, J. Algebra, 284(2) (2005), 679-704.
  • [2] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and A. ´
  • B. Rodr´ıguez Raposo, Weak C-cleft extensions and weak Galois extensions, J.
  • Algebra, 299(1) (2006), 276-293.
  • [3] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and A. ´
  • B. Rodr´ıguez Raposo, Crossed products in weak contexts, Appl. Categ. Structures,
  • 18(3) (2010), 231-258
  • 4] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Cohomology of algebras over weak Hopf algebras, Homology Homotopy Appl.,
  • 16(1) (2014), 341-369.
  • [5] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Crossed products over weak Hopf algebras related to cleft extensions and cohomology,
  • Chin. Ann. Math. Ser. B, 35(2) (2014), 161-190.
  • [6] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Weak Hopf quasigroups, to appear in Asian J. Math., arXiv:1410.2180 (2014).
  • [7] T. Brzezi´nski, On modules associated to coalgebra Galois extensions, J. Algebra,
  • 215(1) (1999), 290-317.
  • [8] S. Caenepeel and E. de Groot, Modules over weak entwining structures, Contemp.
  • Math., 267 (2000), 31-54.
  • [9] Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. Algebra, 17(12)
  • (1989), 3053-3085.
  • [10] Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra,
  • 14(5) (1986), 801-817.
  • [11] J. M. Fern´andez Vilaboa and E. Villanueva Novoa, A characterization of the
  • cleft comodule triples, Comm. Algebra, 16(3) (1988), 613-622.
  • [12] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, SpringerVerlag,
  • New York, 1995.
Year 2017, , 23 - 38, 17.01.2017
https://doi.org/10.24330/ieja.295749

Abstract

References

  • [1] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and ´
  • A. B. Rodr´ıguez Raposo, Weak C-cleft extensions, weak entwining structures
  • and weak Hopf algebras, J. Algebra, 284(2) (2005), 679-704.
  • [2] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and A. ´
  • B. Rodr´ıguez Raposo, Weak C-cleft extensions and weak Galois extensions, J.
  • Algebra, 299(1) (2006), 276-293.
  • [3] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa, R. Gonz´alez Rodr´ıguez and A. ´
  • B. Rodr´ıguez Raposo, Crossed products in weak contexts, Appl. Categ. Structures,
  • 18(3) (2010), 231-258
  • 4] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Cohomology of algebras over weak Hopf algebras, Homology Homotopy Appl.,
  • 16(1) (2014), 341-369.
  • [5] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Crossed products over weak Hopf algebras related to cleft extensions and cohomology,
  • Chin. Ann. Math. Ser. B, 35(2) (2014), 161-190.
  • [6] J. N. Alonso Alvarez, J. M. Fern´andez Vilaboa and R. Gonz´alez Rodr´ıguez, ´
  • Weak Hopf quasigroups, to appear in Asian J. Math., arXiv:1410.2180 (2014).
  • [7] T. Brzezi´nski, On modules associated to coalgebra Galois extensions, J. Algebra,
  • 215(1) (1999), 290-317.
  • [8] S. Caenepeel and E. de Groot, Modules over weak entwining structures, Contemp.
  • Math., 267 (2000), 31-54.
  • [9] Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. Algebra, 17(12)
  • (1989), 3053-3085.
  • [10] Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra,
  • 14(5) (1986), 801-817.
  • [11] J. M. Fern´andez Vilaboa and E. Villanueva Novoa, A characterization of the
  • cleft comodule triples, Comm. Algebra, 16(3) (1988), 613-622.
  • [12] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155, SpringerVerlag,
  • New York, 1995.
There are 29 citations in total.

Details

Journal Section Articles
Authors

J. N. Alonso Alvarez This is me

J. M. Fernandez Vilaboa This is me

R. Gonzalez Rodriıguez This is me

Publication Date January 17, 2017
Published in Issue Year 2017

Cite

APA Alvarez, J. N. A., Vilaboa, J. M. F., & Rodriıguez, R. G. (2017). H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. International Electronic Journal of Algebra, 21(21), 23-38. https://doi.org/10.24330/ieja.295749
AMA Alvarez JNA, Vilaboa JMF, Rodriıguez RG. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. IEJA. January 2017;21(21):23-38. doi:10.24330/ieja.295749
Chicago Alvarez, J. N. Alonso, J. M. Fernandez Vilaboa, and R. Gonzalez Rodriıguez. “H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 23-38. https://doi.org/10.24330/ieja.295749.
EndNote Alvarez JNA, Vilaboa JMF, Rodriıguez RG (January 1, 2017) H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. International Electronic Journal of Algebra 21 21 23–38.
IEEE J. N. A. Alvarez, J. M. F. Vilaboa, and R. G. Rodriıguez, “H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS”, IEJA, vol. 21, no. 21, pp. 23–38, 2017, doi: 10.24330/ieja.295749.
ISNAD Alvarez, J. N. Alonso et al. “H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS”. International Electronic Journal of Algebra 21/21 (January 2017), 23-38. https://doi.org/10.24330/ieja.295749.
JAMA Alvarez JNA, Vilaboa JMF, Rodriıguez RG. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. IEJA. 2017;21:23–38.
MLA Alvarez, J. N. Alonso et al. “H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 23-38, doi:10.24330/ieja.295749.
Vancouver Alvarez JNA, Vilaboa JMF, Rodriıguez RG. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. IEJA. 2017;21(21):23-38.