On osp(2|2)-relative cohomology of the Lie superalgebra of contact vector fields and deformations

We consider the 1 vertical bar 2-dimensional real superspace R-1 vertical bar 2 endowed with its standard contact structure defined by the 1-form alpha. The conformal Lie superalgebra kappa(2) acts on R-1 vertical bar 2 as the Lie superalgebra of contact vector fields; it contains the Mobius superalgebra osp(2 vertical bar 2). We classify osp(1 vertical bar 2)-invariant superskew-symmetric binary differential operators from kappa(2)boolean AND kappa(2) to D-lambda,D-mu vanishing on 051)(2 vertical bar 2), where D-lambda,D-mu is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential osp(1 vertical bar 2)-relative cohomology of kappa(2) with coefficients in D-lambda,D-mu. We study generic formal osp(1 vertical bar 2)-trivial deformations of the kappa(2)-module structure on the direct sum of the superspaces of weighted densities. This work is the simplest superization of a result by Bouarroudj (2007). (C) 2018 Elsevier B.V. All rights reserved.