Skip to main content
SpringerLink
Log in

Crossed products over weak Hopf algebras related to cleft extensions and cohomology

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions A H ↪ A, where A H is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over A H . Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group \(H_{\phi _{Z(A_H )} }^2 \) (H,Z(A H )), where Z(A H ) is the center of A H .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agore, A. and Militaru, G., Extending structures II: The quantum version, J. Algebra, 336, 2011, 321–341.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alonso, Á. J. N., Fernández, V. J. M., González, R. R. and Rodríguez, R. A. B., Weak C-cleft extensions, weak entwining structures and weak Hopf algebras, J. Algebra, 284, 2005, 679–704.

    Article  MATH  MathSciNet  Google Scholar 

  3. Alonso, Á. J. N., Fernández, V. J. M., González, R. R. and Rodríguez, R. A. B., Weak C-cleft extensions and weak Galois extensions, J. Algebra, 299, 2006, 276–293.

    Article  MATH  MathSciNet  Google Scholar 

  4. Alonso, Á. J. N., Fernández, V. J. M., González, R. R. and Rodríguez, R. A. B., Crosssed products in weak contexts, Appl. Cat. Structures, 18, 2010, 231–258.

    Article  MATH  Google Scholar 

  5. Alonso, Á. J. N., Fernández, V. J. M. and González, R. R., Cohomology of algebras over weak Hopf algebras, 2012. arXiv:1206.3850

    Google Scholar 

  6. Blattner, R. J., Cohen, M. and Montgomery, S., Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc., 298, 1986, 671–711.

    Article  MATH  MathSciNet  Google Scholar 

  7. Blattner, R. J. and Montgomery, S., Crossed products and Galois extensions of Hopf algebras, Pacific J. Math., 137, 1989, 37–54.

    Article  MATH  MathSciNet  Google Scholar 

  8. Böhm, G., The weak theory of monads, Advances Math., 225, 2010, 1–32.

    Article  MATH  Google Scholar 

  9. Böhm, G., Nill, F. and Szlachányi, K., Weak Hopf algebras, I: Integral theory and C*-structures, J. Algebra, 221, 1999, 385–438.

    Article  MATH  MathSciNet  Google Scholar 

  10. Brzeziński, T., Crossed products by a coalgebra, Comm. Algebra, 25, 1997, 3551–3575.

    Article  MATH  MathSciNet  Google Scholar 

  11. Bulacu, D., The weak braided Hopf algebra structure of some Cayley-Dickson algebras, J. Algebra, 322, 2009, 2404–2427.

    Article  MATH  MathSciNet  Google Scholar 

  12. Bulacu, D., A Clifford algebra is a weak Hopf algebra in a suitable symmetric monoidal category, J. Algebra, 332, 2011, 244–284.

    Article  MATH  MathSciNet  Google Scholar 

  13. Caenepeel, S., De Groot, E., Modules over weak entwining structures, Contemp. Math., 267, 2000, 4701–4735.

    Google Scholar 

  14. Doi, Y., Equivalent crossed products for a Hopf algebra, Comm. Algebra, 17, 1989, 3053–3085.

    Article  MATH  MathSciNet  Google Scholar 

  15. Doi, Y. and Takeuchi, T., Cleft comodule algebras for a bialgebra, Comm. Algebra, 14, 1986, 801–817.

    MATH  MathSciNet  Google Scholar 

  16. Fernández, V. J. M., González, R. R. and Rodríguez, R. A. B., Preunits and weak crossed products, J. Pure Appl. Algebra, 213, 2009, 2244–2261.

    Article  MATH  MathSciNet  Google Scholar 

  17. Fernández, V. J. M., González, R. R. and Rodríguez, R. A. B., Partial and unified crossed products are weak crossed products, Contemp. Math., 585, 2013, 261–274.

    Article  Google Scholar 

  18. Lack, S. and Street, R., The formal theory of monads II, J. Pure Appl. Algebra, 175, 2002, 243–265.

    Article  MATH  MathSciNet  Google Scholar 

  19. Montgomery, S., Hopf algebras and their actions on rings, CBMS, 82, A. M. S., Providence, RI, 1992.

  20. Muniz, S., Alves, M., Batista, E., et al., Twisted partial actions of Hopf algebras, 2011. arXiv:1111.1281

    Google Scholar 

  21. Nikshych, D. and Vainerman, L., Finite quantum groupoids and their applications, MSRI Publications, 43, 2002, 211–262.

    MathSciNet  Google Scholar 

  22. Rodríguez, R. A. B., Crossed products for weak Hopf algebras, Comm. Algebra, 37, 2009, 2274–2289.

    Article  MATH  MathSciNet  Google Scholar 

  23. Sweedler, M., Cohomology of algebras over Hopf algebras, Trans. American Math. Soc., 133, 1968, 205–239.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de Vigo, Campus Universitario Lagoas-Marcosende, E-36280, Vigo, Spain

    José Nicanor Alonso Álvarez

  2. Departamento de Álxebra, Universidad de Santiago de Compostela, E-15771, Santiago de Compostela, Spain

    José Manuel Fernández Vilaboa

  3. Departamento de Matemática Aplicada II, Universidad de Vigo, Campus Universitario Lagoas-Marcosende, E-36310, Vigo, Spain

    Ramón González Rodríguez

Authors
  1. José Nicanor Alonso Álvarez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Manuel Fernández Vilaboa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ramón González Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Nicanor Alonso Álvarez.

Additional information

This work was supported by the project of Ministerio de Ciencia e Innovación (No.MTM2010-15634) and Fondo Europeo de Desarrollo Regional.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Álvarez, J.N.A., Vilaboa, J.M.F. & Rodríguez, R.G. Crossed products over weak Hopf algebras related to cleft extensions and cohomology. Chin. Ann. Math. Ser. B 35, 161–190 (2014). https://doi.org/10.1007/s11401-014-0828-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-014-0828-x

Keywords

2000 MR Subject Classification

Search

Navigation