Skip to main content
SpringerLink
Log in

Generalizations of the Sweedler Dual

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.

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. Abuhlail, J.Y., Al-Sulaiman, N.: Hopf Semialgebras. Comm. Algebra 43, 1241–1278 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abuhlail, J.Y., Gómez-Torrecillas, J., Wisbauer, R.: Dual coalgebras of algebras over commutative rings. J. Pure Appl. Algebra 153, 107–120 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Adámek, J., Herrlich, H., Strecker, G.E.: Abstract and Concrete Categories. John Wiley, New York (1990)

    MATH  Google Scholar 

  4. Aguiar, M., Mahajan, S.: Monoidal Functors, Species and Hopf Algebras, CRM Monograph Series 29 American Math. Society, Providence, RI (2010)

  5. Anel, M., Joyal, A.: Sweedler theory of (co)algebras and the bar-cobar constructions (2013). arXiv:1309.6952

  6. Chen, C.Y., Nichols, W.D.: A duality theorem for Hopf module algebras over Dedekind rings. Comm. Algebra 18(10), 3209–3221 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Day, B.J.: Construction of biclosed categories. University of New South Wales, PhD thesis (1970)

    MATH  Google Scholar 

  8. Dăscălescu, S., Năstăsescu, C., Raianu, S.: Hopf algebras - An introduction, Pure and Applied Mathematics, vol. 235. Marcel Dekker, New York-Basel (2001)

  9. Hyland, M., López Franco, I, Vasilakopoulou, C.: Hopf measuring comonoids and enrichment. arXiv:1509.07632 (2015)

  10. Kelly, G.M.: Doctrinal adjunction, Lecture Notes in Math, vol. 420, pp 257–280. Springer-Verlag, Berlin, Heidelberg, New York (1974)

  11. Kelly, G.M., Street, R.: Review of the elements of 2-categories, Lecture Notes in Math, vol. 420, pp 75–103. Springer-Verlag, Berlin, Heidelberg, New York (1974)

  12. Mac Lane, S.: : Categories for the Working Mathematician, Second Edition. Springer, New York (1998)

    MATH  Google Scholar 

  13. Pareigis, B.: Non-additive ring and module theory I, vol. 24, pp 189–204 (1977). Debrecen

  14. Poinsot, L., Porst, H.-E.: The dual R-ring of an R-coring revisited. Comm. Algebra 44, 944–964 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Porst, H.-E.: On Categories of Monoids, Comonoids, and Bimonoids. Quaest. Math. 31, 127–139 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Porst, H.-E.: Dual adjunctions between algebras and coalgebras. Arab. J. Sci. Eng. Sect. C 33(2), 407–411 (2008)

    MathSciNet  MATH  Google Scholar 

  17. Porst, H.-E.: Limits and Colimits of Hopf Algebras. J. Algebra 328, 254–267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Porst, H.-E.: The Formal Theory of Hopf Algebras, Part I. Quaest. Math. 38, 631–682 (2015)

    Article  MathSciNet  Google Scholar 

  19. Porst, H.-E.: The Formal Theory of Hopf Algebras, Part II. Quaest. Math. 38, 683–708 (2015)

    Article  MathSciNet  Google Scholar 

  20. Schauenburg, P.: Tannaka Duality for Arbitrary Hopf Algebras, Algebra Berichte, vol. 66. Verlag Reinhard Fischer, München (1992)

  21. Street, R.H.: Quantum Groups: a path to current algebra, Australian Math. Society Lecture Series, vol. 19. Cambridge University Press (2007)

  22. Sweedler, M.E.: Hopf Algebras. Benjamin, New York (1969)

    MATH  Google Scholar 

  23. Sweedler, M: The predual theorem to the Jacobson-Bourbaki theorem. Trans. Amer. Math. Soc. 213, 391–406 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ulmer, F.: Bialgebras in Locally Presentable Categories. Preprint, Universität Wuppertal (1977)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch, South Africa

    Hans-E. Porst

  2. Department of Mathematics, University of Bremen, 28359, Bremen, Germany

    Hans-E. Porst

  3. Centre of Australian Category Theory, Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

    Ross Street

Authors
  1. Hans-E. Porst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ross Street
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans-E. Porst.

Additional information

To the memory of Horst Herrlich

The second author gratefully acknowledges the support of Australian Research Council Discovery Grant DP130101969.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Porst, HE., Street, R. Generalizations of the Sweedler Dual. Appl Categor Struct 24, 619–647 (2016). https://doi.org/10.1007/s10485-016-9450-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9450-2

Keywords

Mathematics Subject Classification (2010)

Search

Navigation