Cohomology of the Lie Superalgebra of Contact Vector Fields on ð,1|1 and Deformations of the Superspace of Symbols

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal osp(1|2)-trivial deformations of the K(1)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal osp(1|2)-trivial deformation of this K(1)-module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On sl(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to osp(N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.