Cohomology of $\mathfrak{osp}(2|2)$ acting on the spaces of linear differential operators on the superspace ${\mathbb{R}}^{1|2}$

Abstract

We compute the first differential cohomology of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2|2)$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1,\,2)-dimensional real superspace. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. This work is a simplest generalization of a result by Basdouri and Ben Ammar [Cohomology of $\mathfrak{osp}(1|2)$ with coefficients in ${\mathfrak{D}}_{\lambda ,\mu }$.