Skip to main content
SpringerLink
Log in

A van Est Isomorphism for Bicrossed Product Hopf Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.

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. Beggs E.J., Gould J.D., Majid S.: Finite group factorizations and braiding. J. Algebra 181(1), 112–151 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beggs E., Majid S.: Quasitriangular and differential structures on bicrossproduct Hopf algebras. J. Algebra 219(2), 682–727 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chevalley C., Eilenberg S.: Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63, 85–124 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  4. Connes A., Moscovici H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198(1), 199–246 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Connes, A., Moscovici, H.: Cyclic cohomology and Hopf algebra symmetry [1800488], Conférence Moshé Flato 1999, Vol. I (Dijon), Math. Phys. Stud., Vol. 21, Dordrecht: Kluwer Acad. Publ., 2000, pp. 121–147

  6. Hadfield T., Majid S.: Bicrossproduct approach to the Connes-Moscovici Hopf algebra. J. Algebra 312(1), 228–256 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y.: Hopf-cyclic homology and cohomology with coefficients. C. R. Math. Acad. Sci. Paris 338(9), 667–672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hajac P.M., Khalkhali M., Rangipour B., Sommerhäuser Y.: Stable anti-Yetter-Drinfeld modules. C. R. Math. Acad. Sci. Paris 338(8), 587–590 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. : On representations of Lie algebras. Ann. of Math. 50(2), 900–915 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hochschild G.: Algebraic Lie algebras and representative functions. Illinois J. Math. 3, 499–523 (1959)

    MathSciNet  MATH  Google Scholar 

  11. Hochschild, G.: Introduction to affine algebraic groups. San Francisco, CA: Holden-Day Inc., 1971

  12. Hochschild G., Mostow G.D.: Representations and representative functions of Lie groups. Ann. of Math. 66(2), 495–542 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hochschild G., Mostow G.D.: On the algebra of repesentative functions of an analytic group. Amer. J. Math. 83, 111–136 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hochschild G.P.: Basic theory of algebraic groups and Lie algebras, Graduate Texts in Mathematics, Vol. 75. Springer-Verlag, New York (1981)

    Book  Google Scholar 

  15. Christian Kassel: Quantum groups, Graduate Texts in Mathematics, Vol. 155. Springer-Verlag, New York (1995)

    Google Scholar 

  16. Khalkhali M., Rangipour B.: Invariant cyclic homology. K-Theory 28(2), 183–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Knapp A.W.: Lie groups, Lie algebras, and cohomology, Mathematical Notes, Vol. 34. Princeton University Press, Princeton, NJ (1988)

    Google Scholar 

  18. Kumar, S., Neeb, K.-H.: Extensions of algebraic groups, Studies in Lie theory, Progr. Math., Vol. 243, Boston, MA: Birkhäuser Boston, 2006, pp. 365–376

  19. Majid S., Ruegg H.: Bicrossproduct structure of κ-Poincaré group and non-commutative geometry. Phys. Lett. B 334(3-4), 348–354 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Majid S.: Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations. Pacific J. Math. 141(2), 311–332 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Majid S.: Physics for algebraists: noncommutative and noncocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130(1), 17–64 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Majid S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  23. Moscovici H., Rangipour B.: Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1. Adv. Math. 210(1), 323–374 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Moscovici H., Rangipour B.: Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology. Adv. Math. 220(3), 706–790 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Moscovici H., Rangipour B.: Hopf cyclic cohomology and transverse characteristic classes. Adv. Math. 227(1), 654–729 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rangipour, B., Sütlü, S.: Lie-Hopf algebras and their Hopf-cyclic cohomology. http://arxiv.org/abs/1012.482v1 [math.OA], 2010

  27. Rangipour, B., Sütlü, S.: Cyclic cohomology of Lie algebras, http://arxiv.org/abs/110802806v1 [math. OA], 2011

  28. Rangipour, B., Sütlü, S.: SAYD modules over Lie-Hopf algebras, http://arxiv.org/abs/1108.6101v2 [math. OA], 2011

  29. Takeuchi M.: Matched pairs of groups and bismash products of Hopf algebras. Comm. Algebra 9(8), 841–882 (1981)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, Canada

    Bahram Rangipour & Serkan Sütlü

Authors
  1. Bahram Rangipour
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Serkan Sütlü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bahram Rangipour.

Additional information

Communicated by A. Connes

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rangipour, B., Sütlü, S. A van Est Isomorphism for Bicrossed Product Hopf Algebras. Commun. Math. Phys. 311, 491–511 (2012). https://doi.org/10.1007/s00220-012-1452-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1452-9

Keywords

Search

Navigation