Abstract
Gelfand and Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bott, R. and Segal, G.: The cohomology of the vector fields on a manifold, Topology 16 (1977), 285-298.
Fuks, D. B.: Cohomology of Infinite Dimensional Lie Algebras, Consultant Bureau, New York, 1986.
Gelfand, I. M. and Fuks, D. B.: Cohomologies of Lie algebra of tangential vector fields of a smooth manifold, Funct. Anal. Appl. 3 (1969), 194-210.
Haefliger, A.: Sur la cohomologie de l'algèbre de Lie des champs de vecteurs, Ann. Sci. Ecole Norm. Sup. (4) 9 (1976), 503-532.
Loday, J.-L.: Cup-product for Leibniz cohomology and dual Leibniz algebras, Math. Scand. 77(2) (1995), 189-196.
Loday, J.-L. and Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), 569-572.
Livernet, M.: Rational homotopy of Leibniz algebras, Manuscripta Math. 96 (1998), 295-315.
Lodder, J.: Leibniz cohomology for differentiable manifolds. Ann. Inst. Fourier (Grenoble) 48(1) (1998) 73-95.
Oudom, J.-M.: Diagonale et cup-produit en cohomologie des algèbres de Leibniz, Comm. Algebra 24(11) (1996), 3623-3635.
Pirashvili, T.: On Leibniz homology, Ann. Inst. Fourier (Grenoble) 44(2) (1994), 401-411.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Frabetti, A., Wagemann, F. On the Leibniz Cohomology of Vector Fields. Annals of Global Analysis and Geometry 21, 177–190 (2002). https://doi.org/10.1023/A:1014740320370
Issue Date:
DOI: https://doi.org/10.1023/A:1014740320370