The Binary $\mathfrak{aff}(n|1)$-Invariant Differential Operators On Weighted Densities On The Superspace $\mathbb{R}^{1|n}$ And $\mathfrak{aff}(n|1)$-Relative Cohomology

Abstract: We consider the $\mathfrak{aff}(n|1)-$module structure on the spaces of differential bilinear operators acting on the superspaces of weighted densities. We classify $\mathfrak{aff}(n|1)-$invariant binary differential operators acting on the spaces of weighted densities. This result allows us to compute the first $\mathfrak{aff}(n|1)-$relative differential cohomology of $\mathcal{K}(n)$ with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities.