Communications in Mathematical Physics

, Volume 146, Issue 2, pp 343–356 | Cite as

Cyclic homology of differential operators, the Virasoro algebra and aq-analogue

  • Christian Kassel


We show how methods from cyclic homology give easily an explicit 2-cocycle ϕ on the Lie algebra of differential operators of the circle such that ϕ restricts to the cocycle defining the Virasoro algebra. The same methods yield also aq-analogue of ϕ as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity. We use an algebra ofq-difference operators andq-analogues of Koszul and the Rham complexes to construct these “quantum” cocycles.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Operator 
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  1. 1.
    Bourbaki, N.: Algèbre, Chap. 10. Algèbre homologique. Paris: Masson 1980Google Scholar
  2. 2.
    Brylinski, J.-L.: Some examples of Hochschild and cyclic homology. Lecture Notes in Mathematics, vol 1271, pp. 33–72. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  3. 3.
    Brylinski, J.-L., Getzler, E.: The homology of algebras of pseudo-differential symbols and the non-commutative residue.K-Theory1, 385–403 (1987)CrossRefGoogle Scholar
  4. 4.
    Getzler, E.: Cyclic homology and the Beilinson-Manin-Schechtman central extension. Proc. A.M.S.104, 729–734 (1988)Google Scholar
  5. 5.
    Kac, V.G., Peterson, D.H.: Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. Natl. Acad. Sci. USA78, 3308–3312 (1981)Google Scholar
  6. 6.
    Takhtadjian, L.A.: Noncommutative homology of quantum tori. Funkts. Anal. Pril.23, 75–76 (1989); English translation: Funct. Anal. Appl.23, 147–149 (1989)Google Scholar
  7. 7.
    Wodzicki, M.: Cyclic homology of differential operators. Duke Math. J.54, 641–647 (1987)Google Scholar
  8. 8.
    Beilinson, A.A., Schechtman, V.V.: Determinant bundles and Virasoro algebras. Commun. Math. Phys.118, 651–701 (1988)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Christian Kassel
    • 1
  1. 1.Institut de Recherche Mathématique AvancéeUniversité Louis Pasteur-C.N.R.S.StrasbourgFrance

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