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

Abstract

We consider the spaces $\mathcal{F}_μ$ of polynomial $μ$-densities on the line as $\mathfrak{sl}(2)$-modules and then we compute the cohomological spaces $\mathrm{H}^1_\mathrm{diff}(\mathfrak{sl}(2), \mathcal{D}_{\barλ,μ})$, where $μ\in \mathbb{R}$, $\barλ=(λ_1,\dots,λ_n) \in\mathbb{R}^n$ and $\mathcal{D}_{\barλ,μ}$ is the space of $n$-ary differential operators from $\mathcal{F}_{λ_1}\otimes\cdots\otimes \mathcal{F}_{λ_n}$ to $\mathcal{F}_μ$.