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

We consider the 1|2-dimensional real superspace R1|2 endowed with its standard contact structure defined by the 1-form α. The conformal Lie superalgebra K(2) acts on R1|2 as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(2|2). We classify osp(2|2)-invariant superskew-symmetric binary differential operators from K(2)∧K(2) to Dλ,μ vanishing on osp(2|2), where Dλ,μ is the superspace of linear differential operators between the superspaces of weighted densities. This result allows us to compute the second differential osp(2|2)-relative cohomology of K(2) with coefficients in Dλ,μ. We study generic formal osp(2|2)-trivial deformations of the K(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).