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