H-GALOIS EXTENSIONS WITH NORMAL BASIS FOR WEAK HOPF ALGEBRAS

Year 2017, , 23 - 38, 17.01.2017

J. N. Alonso Alvarez J. M. Fernandez Vilaboa R. Gonzalez Rodriıguez

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 . 

Keywords

H-Galois extensions, normal basis

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.