Cohomology of $\mathfrak{sl}(2)$ 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 ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{sl}(2),{\mathcal{D}}_{\bar{\lambda },\mu })$, 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 ${\mathcal{F}}_{\mu }$.