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H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

Year 2017, Volume 21, Issue 21, 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, Volume 21, Issue 21, 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.

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, Volume 21, Issue 21

Cite

Bibtex @research article { ieja295749, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {23 - 38}, doi = {10.24330/ieja.295749}, title = {H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS}, key = {cite}, author = {Alvarez, J. N. Alonso and Vilaboa, J. M. Fernandez and Rodriıguez, R. Gonzalez} }
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 . DOI: 10.24330/ieja.295749
MLA Alvarez, J. N. A. , Vilaboa, J. M. F. , Rodriıguez, R. G. "H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS" . International Electronic Journal of Algebra 21 (2017 ): 23-38 <https://dergipark.org.tr/en/pub/ieja/issue/27921/295749>
Chicago Alvarez, J. N. A. , Vilaboa, J. M. F. , Rodriıguez, R. G. "H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS". International Electronic Journal of Algebra 21 (2017 ): 23-38
RIS TY - JOUR T1 - H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS AU - J. N. AlonsoAlvarez, J. M. FernandezVilaboa, R. GonzalezRodriıguez Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.295749 DO - 10.24330/ieja.295749 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 23 EP - 38 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.295749 UR - https://doi.org/10.24330/ieja.295749 Y2 - 2016 ER -
EndNote %0 International Electronic Journal of Algebra H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS %A J. N. Alonso Alvarez , J. M. Fernandez Vilaboa , R. Gonzalez Rodriıguez %T H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.295749 %U 10.24330/ieja.295749
ISNAD Alvarez, J. N. Alonso , Vilaboa, J. M. Fernandez , Rodriıguez, R. Gonzalez . "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
AMA Alvarez J. N. A. , Vilaboa J. M. F. , Rodriıguez R. G. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. IEJA. 2017; 21(21): 23-38.
Vancouver Alvarez J. N. A. , Vilaboa J. M. F. , Rodriıguez R. G. H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS. International Electronic Journal of Algebra. 2017; 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", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 23-38, Jan. 2017, doi:10.24330/ieja.295749