On the cohomology of the Lie Superalgebra of contact vector fields on S1/1
Département de Mathématiques, Faculté des Sciences de Sfax, route de Soukra, 3018 Sfax BP 802, Tunisie, E-mail address : bagreba@fss.rnu.tn
Institut Supérieur de Sciences Appliquées et Technologie, Sousse, E-mail address: benfraj_nizar@yahoo.fr
Abstract
We investigate the first cohomology space attached to the embedding of the Lie superalgebra K(l) of contact vector fields on the supercircle S1/1 in the Lie Superalgebra of superpseudodifferential operators. Following the paper of V. Ovsienko and C. Roger, Deforming the Lie algebra of vector fields on SI inside the Poisson algebra on T*S1, Comm. Math. Phys., 198 (1998) 97-110, we show that this space is four-dimensional with only even cocycles and we calculate explicitly four l-cocycles representing non-trivial generating cohomology classes.