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 ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}(1),{\mathrm{D}}_{\lambda ,\mu })$ 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 ${\mathrm{H}}_{\mathrm{diff}}^1(\mathfrak{aff}(1|1),{\mathfrak{D}}_{\lambda ,\mu })$.

14 pages. arXiv admin note: text overlap with arXiv:0911.2769, arXiv:1512.02847