Cohomology of $\mathfrak {aff}(1)$ and $\mathfrak {aff}(1|1)$ acting on the space of $n$-ary differential operators on the superspace $\mathbb{R}^{1|1}$

Abstract: We consider the $\mu$-densities spaces $\mathcal{F}\mu$ with $\mu\in\mathbb{R}$, we compute the space $\mathrm{H}^1\mathrm{diff}(\mathfrak{aff}(1),\mathrm{D}{\lambda,\mu})$ where $\lambda=(\lambda_1,\dots,\lambda_n)\in\mathbb{R}^n$ and $\mathrm{D}{\lambda,\mu}$ is the space of $n$-ary differential operators from $\mathcal{F}{\lambda_1}\otimes\cdots\otimes\mathcal{F}{\lambda_n}$ to $\mathcal{F}\mu$. We also compute the super analog space $\mathrm{H}^1\mathrm{diff}(\mathfrak{aff}(1|1),\mathfrak{D}_{\lambda,\mu})$.