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

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.