Advertisement

Frobenius and Separable Functors for the Category of Entwined Modules over Cowreaths, I: General Theory

  • D. Bulacu
  • S. Caenepeel
  • B. TorrecillasEmail author
Article
  • 5 Downloads

Abstract

Entwined modules over cowreaths in a monoidal category are introduced. They can be identified to coalgebras in an appropriate monoidal category. It is investigated when such coalgebras are Frobenius (resp. separable), and when the forgetful functor from entwined modules to representations of the underlying algebra is Frobenius (resp. separable). These properties are equivalent when the unit object of the category is a ⊗-generator.

Keywords

Module category Cowreath Entwined module Frobenius functor Separable functor Frobenius coalgebra Coseparable coalgebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors are very grateful to the referee for careful reading of the paper and valuable suggestions and comments.

References

  1. 1.
    Borceux, F.: Handbook of Categorical Algebra II: Categories and Structures: Encyclopedia Math Appl, vol. 51. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  2. 2.
    Brzezinski, T., Majid, S.: Coalgebra bundles. Comm. Math. Phys. 191, 467–492 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brzezinski, T.: On modules associated to coalgebra-Galois extensions. J. Algebra 215, 290–317 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brzezinski, T.: Frobenius properties and Maschke-type theorems for entwined modules. Proc. Amer. Math. Soc. 128, 2261–2270 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bulacu, D., Caenepeel, S.: Corings in monoidal categories. In: New Techniques in Hopf Algebras and Graded Ring Theory, pp 53–78. K. Vlaam. Acad. België Wet. Kunsten (KVAB), Brussels (2007)Google Scholar
  6. 6.
    Bulacu, D., Caenepeel, S.: Monoidal ring and coring structures obtained from wreaths and cowreaths. Algebr. Represent. Theory 17, 1035–1082 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bulacu, D., Caenepeel, S., Torrecillas, B.: Frobenius and separable functors for the category of entwined modules over cowreaths, II: Applications. J. Algebra 515, 236–277 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bulacu, D., Torrecillas, B.: On Doi-Hopf modules and Yetter-Drinfeld modules in symmetric monoidal categories. Bull. Belg. Math. Soc. - Simon Stevin 21, 89–115 (2014)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bulacu, D., Torrecillas, B.: On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths. J. Noncommut. Geom. 9, 707–774 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caenepeel, S., De Lombaerde, M.: A categorical approach to Turaev’s Hopf group-coalgebras. Comm. Algebra 34, 2631–2657 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Caenepeel, S., Militaru, G., Ion, B., Zhu, S.: Separable functors for the category of Doi-Hopf modules. Appl. Adv. Math. 145, 239–290 (1999)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Caenepeel, S., Militaru, G., Zhu, S.: Crossed modules and Doi-Hopf modules. Israel J. Math. 100, 221–247 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caenepeel, S., Militaru, G., Shenglin, Zhu: A Maschke-type theorem for Doi-Hopf modules. Appl. J. Algebra 187, 388–412 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Caenepeel, S., Militaru, G., Zhu, S.: Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties. Trans. Amer. Math. Soc. 349, 4311–4342 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hobst, D., Pareigis, B.: Double quantum groups. J. Algebra 242, 460–494 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Johnstone, P.T.: Topos Theory L.M.S. Monographs, vol. 10. Academic Press, London (1977)Google Scholar
  17. 17.
    El Kaoutit, L.: Extended distributive law: Cowreaths over corings. J. Algebra Appl. 09, 135–171 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kassel, C: Quantum Groups: Graduate Texts in Mathematics, vol. 155. Springer, Berlin (1995)CrossRefGoogle Scholar
  19. 19.
    Lack, S., Street, R.: The formal theory of monads II. J. Pure Appl. Algebra 175, 243–265 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Larson, R.G.: Coseparable Hopf algebras. J. Pure Appl. Algebra 3, 261–267 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Grad Texts Math, vol. 5. Springer, Berlin (1998)Google Scholar
  22. 22.
    Nastasescu, C., Van den Bergh, M., Van Oystaeyen, F.: Separable functors applied to graded rings. J. Algebra 123, 397–413 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pareigis, B.: Non-additive ring and module theory V. Projective and coflat objects. Algebra Ber., 40 (1980)Google Scholar
  24. 24.
    Rafael, D.M.: Separable functors revisited. Comm. Algebra 18, 1445–1459 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Schauenburg, P.: Actions on monoidal categories and generalized Hopf smash products. J. Algebra 270, 521–563 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Street, R.: Frobenius monads and pseudomonoids. J. Math. Phys. 45, 3930–3948 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Street, R.: Wreaths, mixed wreaths and twisted conditions. Tbilisi Math. J. 10, 1–22 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tambara, D.: The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 425–456 (1990)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Wolff, H: V-cat AND V-graph. J. Pure Appl. Algebra 4, 123–135 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania
  2. 2.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  3. 3.Department of Algebra and AnalysisUniversidad de AlmeríaAlmeríaSpain

Personalised recommendations