Skip to main content
SpringerLink
Log in

Duality Features of Left Hopf Algebroids

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

Abstract

We explore special features of the pair (U ∗,U ∗) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S ∗ from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phùng’s categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.

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. Anderson, F., Fuller, K.: Rings and Categories of Modules. 2nd edn., Graduate Texts in Mathematics, vol. 13. Springer, New York (1992)

  2. Beck, J.: Distributive Laws Sem.on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), pp. 119–140. Springer, Berlin (1969)

    Book  Google Scholar 

  3. Böhm, G.: Hopf algebroids. In: Handbook of Algebra, vol. 6, pp. 173–236. North-Holland, Amsterdam (2009)

  4. Böhm, G., Szlachányi, K.: Hopf algebroids with bijective antipodes: axioms, integrals, and duals. J. Algebra 274(2), 708–750 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borel, A., Grivel, P.-P., Kaup, B., Haefliger, A., Malgrange, B., Ehlers, F.: Algebraic D-modules. Perspectives in Mathematics, vol. 2. Academic, Boston (1987)

  6. Calaque, D., Van den Bergh, M.: Hochschild cohomology and Atiyah classes. Adv. Math. 224, 1839–1889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cartier, P.: Cohomologie des coalgèbres, exposés 4, 5, Séminaire Sophus Lie, tome 2 (1955–1956) Faculté des Sciences de Paris (1957)

  8. Chemla, S.: Poincaré duality for k-A Lie superalgebras. Bull. Soc. Math. France 122(3), 371–397 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Chemla, S.: Duality properties for quantum groups. Pacific J. Math. 252(2), 313–341 (2011) (a more detailed version is available at arXiv:0911.2860)

  10. Chemla, S.: Rigid dualizing complex for quantum enveloping algebras and algebras of generalized differential operators. J. Algebra 276, 80–102 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chemla, S., Gavarini, F.: Duality functors for quantum groupoids. J. Noncomm. Geom. 9(2), 287–358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Huebschmann, J.: Duality for Lie-Rinehart algebras and the modular class. J. Reine Angew. Math. 510, 103–159 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Kadison, L., Szlachányi, K.: Bialgebroid actions on depth two extensions and duality. Adv. Math. 179(1), 75–121 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kashiwara, M.: D-Modules and Microlocal Calculus. Translations of Mathematical Monographs, vol. 217. American Mathematical Society, Providence (2003)

  15. Kowalzig, N.: Hopf algebroids and their cyclic theory. Ph.D. thesis Universiteit Utrecht and Universiteit van Amsterdam (2009)

  16. Kowalzig, N., Krähmer, U.: Duality and products in algebraic (co)homology theories. J. Algebra 323(7), 2063–2081 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kowalzig, N., Posthuma, H.: The cyclic theory of Hopf algebroids. J. Noncomm. Geom. 5(3), 423–476 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Phùng, H.H.: Tannaka-Krein duality for Hopf algebroids. Israel J. Math. 167, 193–225 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rinehart, G.: Differential forms on general commutative algebras. Trans. Amer. Math. Soc. 108, 195–222 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rotman, J.: An Introduction to Homological Algebra. 2nd edn. Universitext, Springer, New York (2009)

  21. Schauenburg, P.: Duals and doubles of quantum groupoids (× R -Hopf algebras) New trends in Hopf algebra theory (La Falda, 1999), Contemp. Math, Amer. Math. Soc., Providence, vol. 267, pp. 273–299 (2000)

  22. Schauenburg, P.: The dual and the double of a Hopf algebroid are Hopf algebroids. arXiv:1504.05057 (2015)

  23. Schneiders, J.-P.: An introduction to \(\mathcal {D}\)-modules. Bull. Soc. Roy. Sci. Liège 63(3–4), 223–295 (1994). Algebraic Analysis Meeting (Liège, 1993)

    MathSciNet  MATH  Google Scholar 

  24. Sweedler, M.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin, New York (1969)

    MATH  Google Scholar 

  25. Takeuchi, M.: Groups of algebras over \(A\otimes \overline {A}\). J. Math. Soc. Japan 29(3), 459–492 (1977)

  26. Xu, P.: Quantum groupoids. Comm. Math. Phys. 216(3), 539–581 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie, 75005, Paris, France

    Sophie Chemla

  2. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133, Roma, Italia

    Fabio Gavarini

  3. Istituto Nazionale di Alta Matematica, P.le Aldo Moro 5, 00185, Roma, Italia

    Niels Kowalzig

Authors
  1. Sophie Chemla
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabio Gavarini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Niels Kowalzig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabio Gavarini.

Additional information

Presented by Susan Montgomery.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chemla, S., Gavarini, F. & Kowalzig, N. Duality Features of Left Hopf Algebroids. Algebr Represent Theor 19, 913–941 (2016). https://doi.org/10.1007/s10468-016-9604-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-016-9604-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation