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

TL;DR

This paper computes the first differential cohomology of the Lie superalgebra $rak{osp}(2|2)$ acting on linear differential operators on weighted densities in a superspace, extending previous results for $rak{osp}(1|2)$.

Contribution

It provides explicit calculations of the first cohomology groups and 1-cocycles for $rak{osp}(2|2)$, generalizing known results for $rak{osp}(1|2)$.

Findings

Explicit 1-cocycles spanning the cohomology groups.

First cohomology computation for $rak{osp}(2|2)$ on differential operators.

Extension of previous $rak{osp}(1|2)$ results.

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 $\frak {osp}(1|2)$ with coefficients in $\frak{D}_{\lambda,\mu}$.