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Crossed Products in Weak Contexts

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Abstract

We define the general notion of crossed products in a weak context, which generalizes the ones defined by Blattner, Cohen and Montgomery, Doi and Takeuchi in the context of Hopf algebras and the one given by Brzeziński. Also, the crossed products obtained by the authors, for weak Hopf algebras living in a symmetric monoidal category and weak C-cleft extensions associated to weak entwined structures, are particular instances of this theory.

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References

  1. Alonso Álvarez, J.N., Fernández Vilaboa, J.M.: Cleft extensions in braided categories. Comm. Algebra 7, 3185–3196 (2000)

    Article  Google Scholar 

  2. Alonso Álvarez, J.N., González Rodríguez, R., Fernández Vilaboa, J.M., López López, M.P., Villanueva Novoa, E.: Weak hopf algebras with projection and weak smash Bialgebra structures. J. Algebra 269(2), 701–725 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alonso Álvarez, J.N., González Rodríguez, R.: Crossed products for weak Hopf algebras with coalgebra splitting. J. Algebra 281, 731–752 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R., Rodríguez Raposo, A.B.: Weak C-cleft extensions, weak entwining structures and weak Hopf algebras. J. Algebra 284, 679–704 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alonso Álvarez, J.N., Fernández Vilaboa, R.J.M., González Rodríguez, R., Rodríguez Raposo, A.B.; Weak C-cleft extensions and weak Galois extensions. J. Algebra 299, 276–293 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bespalov, Y.: Crossed modules and quantum groups in braided categories. Appl. Categ. Structures 5, 155–204 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Blattner, R., Cohen, M., Montgomery, S.: Crossed products and inner actions of Hopf algebras. Trans. Amer. Math. Soc. 298(2), 671–711 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  8. Brzeziński T.: Crossed products by a coalgebra. Comm. Algebra 25(11), 3551–3575 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brzeziński, T.: On modules associated to coalgebra Galois extensions. J. Algebra 215, 290–317 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brzeziński, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191, 467–492 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Böhm, G., Nill, F., Szlachányi, K.: Weak hopf algebras, I. integral theory and C ∗-structure. J. Algebra 221(2), 385–438 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Böhm, G., Szlachányi, K.: Weak hopf algebras, II. Representation theory, dimensions and the markov trace. J. Algebra 233, 156–212 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caenepeel, S., de Groot, E.: Modules over weak entwining structures. Contemp. Math. 267, 31–54 (2000)

    Google Scholar 

  14. Cap, A., Schichl, H., Vanzura, J.: On twisted tensor products of algebras. Comm. Algebra 23, 4701–4735 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Doi, Y.: Cleft comodule algebras and Hopf modules. Comm. Algebra 12, 1115–1169 (1984)

    Article  MathSciNet  Google Scholar 

  16. Doi, Y., Takeuchi, M.: Cleft comodule algebras for a bialgebra. Comm. Algebra 14, 801–817 (1986)

    MATH  MathSciNet  Google Scholar 

  17. Fernández Vilaboa, J.M., López López, M.P.: Cleft comodule triples in a symmetric closed category. Bull. Soc. Math. Belg. 40, 283–297 (1988)

    MATH  Google Scholar 

  18. Guccione, J.A., Guccione, J.J.: Theory of braided Hopf crossed products. J. Algebra 261, 54–101 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kreimer, H.F., Takeuchi, M.: Hopf algebras and Galois extensions of an algebra. Indiana Univ. Math. J. 30, 675–691 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lack, S., Street, R.: The formal theory of monads II. J. Pure Appl. Algebra 175, 243–265 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Majid, S.: Crossed products by braided groups and bosonization. J. Algebra 163, 165–190 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  22. Masuoka, A., Doi, Y.: Generalization of cleft comodule algebras. Comm. Algebra 20, 3703–3721 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nikshych, D., Vainerman, L.: Finite Quantum Groupoids and their applications. New Directions in Hopf Algebras, vol. 43, pp. 211–262. MSRI Publications (2002)

  24. Nikshych, D., Turaev, V., Vainerman, L.: Invariants of knots and 3-manifolds from quantum groupoids. Topology Appl. 127, 91–123 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pastro, C., Street, R.: Weak Hopf monoids in braided monoidal categories. (2008) (arXiv:mathQA/0801.4067)

  26. Tambara, D.: The endomorphism bialgebra of an algebra. J. Fac. Univ. Tokyo Sect. IA 37, 425–456 (1990)

    MATH  MathSciNet  Google Scholar 

  27. Wisbauer, R.: Algebras vs coalgebras. Appl. Categ. Structures 16, 255–295 (2008)

    Google Scholar 

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Authors and Affiliations

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

    J. N. Alonso Álvarez

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

    J. M. Fernández Vilaboa & A. B. Rodríguez Raposo

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

    R. González Rodríguez

Authors
  1. J. N. Alonso Álvarez
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  2. J. M. Fernández Vilaboa
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  3. R. González Rodríguez
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  4. A. B. Rodríguez Raposo
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Corresponding author

Correspondence to R. González Rodríguez.

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Alonso Álvarez, J.N., Fernández Vilaboa, J.M., González Rodríguez, R. et al. Crossed Products in Weak Contexts. Appl Categor Struct 18, 231–258 (2010). https://doi.org/10.1007/s10485-008-9139-2

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  • DOI: https://doi.org/10.1007/s10485-008-9139-2

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