Communications in Mathematical Physics

, Volume 316, Issue 1, pp 199–236 | Cite as

SAYD Modules over Lie-Hopf Algebras



In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.


Hopf Algebra Spectral Sequence Matched Pair Cohomology Class Cyclic Cohomology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Connes A., Moscovici H.: Hopf algebras, cyclic cohomology and the transverse index theorem. Commun. Math. Phys. 198(1), 199–246 (1998)MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Fuks, D. B.: Cohomology of infinite-dimensional Lie algebras. Contemporary Soviet Mathematics, New York: Consultants Bureau 1986, translated from the Russian by A. B. SosinskiĭGoogle Scholar
  3. 3.
    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)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Hochschild G.: Lie algebra cohomology and affine algebraic groups. Illinois J. Math. 18, 170–176 (1974)MathSciNetMATHGoogle Scholar
  6. 6.
    Hochschild G.P.: Basic theory of algebraic groups and Lie algebras. Graduate Texts in Mathematics, Vol. 75. Springer-Verlag, New York (1981)CrossRefGoogle Scholar
  7. 7.
    Jara P., Ştefan D.: Hopf-cyclic homology and relative cyclic homology of Hopf-Galois extensions. Proc. London Math. Soc. (3) 93(1), 138–174 (2006)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Majid S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)MATHCrossRefGoogle Scholar
  9. 9.
    Moscovici H., Rangipour B.: Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1. Adv. Math. 210(1), 323–374 (2007)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Moscovici H., Rangipour B.: Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology. Adv. Math. 220(3), 706–790 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Rangipour B., Sütlü S.: A van Est isomorphism for bicrossed product Hopf algebras. Commun. Math. Phys. 311(2), 491–511 (2012)ADSMATHCrossRefGoogle Scholar
  12. 12.
    Rangipour, B., Sütlü, S.: Cyclic cohomology of Lie algebras. [math.QA], 2011
  13. 13.
    Staic, M. D.: A note on anti-Yetter-Drinfeld modules. Im: Hopf algebras and generalizations, Contemp. Math., Vol. 441, Providence, RI: Amer. Math. Soc., 2007, pp. 149–153Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

Personalised recommendations