Cohomology of $\frak {sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$

Abstract

We consider the spaces $\mathcal{F}_\mu$ of polynomial $\mu$-densities on the line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological spaces $\mathrm{H}^1_\mathrm{diff}(\mathfrak{sl}(2), \mathcal{D}_{\bar{\lambda},\mu})$, where $\mu\in \mathbb{R}$, $\bar{\lambda}=(\lambda_1,\dots,\lambda_n) \in\mathbb{R}^n$ and $\mathcal{D}_{\bar{\lambda},\mu}$ is the space of $n$-ary differential operators from $\mathcal{F}_{\lambda_1}\otimes\cdots\otimes \mathcal{F}_{\lambda_n}$ to $\mathcal{F}_\mu$.