Skip to main content
SpringerLink
Log in

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

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

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.

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. Borceux, F.: Handbook of Categorical Algebra II: Categories and Structures: Encyclopedia Math Appl, vol. 51. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Brzezinski, T.: Frobenius properties and Maschke-type theorems for entwined modules. Proc. Amer. Math. Soc. 128, 2261–2270 (1999)

    Article  MathSciNet  Google Scholar 

  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)

  6. Bulacu, D., Caenepeel, S.: Monoidal ring and coring structures obtained from wreaths and cowreaths. Algebr. Represent. Theory 17, 1035–1082 (2014)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. Bulacu, D., Torrecillas, B.: On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths. J. Noncommut. Geom. 9, 707–774 (2015)

    Article  MathSciNet  Google Scholar 

  10. Caenepeel, S., De Lombaerde, M.: A categorical approach to Turaev’s Hopf group-coalgebras. Comm. Algebra 34, 2631–2657 (2006)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Caenepeel, S., Militaru, G., Zhu, S.: Crossed modules and Doi-Hopf modules. Israel J. Math. 100, 221–247 (1997)

    Article  MathSciNet  Google Scholar 

  13. Caenepeel, S., Militaru, G., Shenglin, Zhu: A Maschke-type theorem for Doi-Hopf modules. Appl. J. Algebra 187, 388–412 (1997)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Hobst, D., Pareigis, B.: Double quantum groups. J. Algebra 242, 460–494 (2001)

    Article  MathSciNet  Google Scholar 

  16. Johnstone, P.T.: Topos Theory L.M.S. Monographs, vol. 10. Academic Press, London (1977)

    Google Scholar 

  17. El Kaoutit, L.: Extended distributive law: Cowreaths over corings. J. Algebra Appl. 09, 135–171 (2010)

    Article  MathSciNet  Google Scholar 

  18. Kassel, C: Quantum Groups: Graduate Texts in Mathematics, vol. 155. Springer, Berlin (1995)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Larson, R.G.: Coseparable Hopf algebras. J. Pure Appl. Algebra 3, 261–267 (1973)

    Article  MathSciNet  Google Scholar 

  21. Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Grad Texts Math, vol. 5. Springer, Berlin (1998)

    Google Scholar 

  22. Nastasescu, C., Van den Bergh, M., Van Oystaeyen, F.: Separable functors applied to graded rings. J. Algebra 123, 397–413 (1989)

    Article  MathSciNet  Google Scholar 

  23. Pareigis, B.: Non-additive ring and module theory V. Projective and coflat objects. Algebra Ber., 40 (1980)

  24. Rafael, D.M.: Separable functors revisited. Comm. Algebra 18, 1445–1459 (1990)

    Article  MathSciNet  Google Scholar 

  25. Schauenburg, P.: Actions on monoidal categories and generalized Hopf smash products. J. Algebra 270, 521–563 (2003)

    Article  MathSciNet  Google Scholar 

  26. Street, R.: The formal theory of monads. J. Pure Appl. Algebra 2, 149–168 (1972)

    Article  MathSciNet  Google Scholar 

  27. Street, R.: Frobenius monads and pseudomonoids. J. Math. Phys. 45, 3930–3948 (2004)

    Article  MathSciNet  Google Scholar 

  28. Street, R.: Wreaths, mixed wreaths and twisted conditions. Tbilisi Math. J. 10, 1–22 (2017)

    Article  MathSciNet  Google Scholar 

  29. Tambara, D.: The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37, 425–456 (1990)

    MathSciNet  MATH  Google Scholar 

  30. Wolff, H: V-cat AND V-graph. J. Pure Appl. Algebra 4, 123–135 (1974)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

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

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014, Bucharest 1, Romania

    D. Bulacu

  2. Faculty of Engineering, Vrije Universiteit Brussel, B-1050, Brussels, Belgium

    S. Caenepeel

  3. Department of Algebra and Analysis, Universidad de Almería, E-04071, Almería, Spain

    B. Torrecillas

Authors
  1. D. Bulacu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Caenepeel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Torrecillas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Torrecillas.

Additional information

Presented by: Steffen Koenig

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author was supported by the UEFISCDI Grant PN-II-ID-PCE-2011- 3-0635, contract no. 253/5.10.2011 of CNCSIS. The second author was supported by research project G.0117.10 “Equivariant Brauer groups and Galois deformations” from FWO-Vlaanderen. The third author was partially supported by FQM 211 from Junta Andalucía and by research project MTM2017-86987 from MEC. The first author thanks the Vrije Universiteit Brussel and the Universidad de Almería for their support and warm hospitality. The authors also thank Bodo Pareigis for sharing his “diagrams” program.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bulacu, D., Caenepeel, S. & Torrecillas, B. Frobenius and Separable Functors for the Category of Entwined Modules over Cowreaths, I: General Theory. Algebr Represent Theor 23, 1119–1157 (2020). https://doi.org/10.1007/s10468-019-09884-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-019-09884-6

Keywords

Search

Navigation