Cohomology of $\mathfrak {aff}(1)$ and $\mathfrak {aff}(1|1)$ acting on the space of $n$-ary differential operators on the superspace $\mathbb{R}^{1|1}$

TL;DR

This paper computes the first cohomology spaces of the affine Lie algebra and its superalgebra acting on spaces of n-ary differential operators on the superspace (1|1), revealing their structure and differences.

Contribution

It provides explicit calculations of the first cohomology groups for (1) and (1|1) acting on n-ary differential operators, extending previous work to the super setting.

Findings

Computed (1) cohomology space ^1_ ext{diff}((1), D_{,})

Computed super analog cohomology space ^1_ ext{diff}((1|1), )

Extended understanding of the cohomological structure of differential operators in superspace contexts

Abstract

We consider the $\mu$-densities spaces $\mathcal{F}_\mu$ with $\mu\in\mathbb{R}$, we compute the space $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1),\mathrm{D}_{\lambda,\mu})$ 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 $\mathrm{H}^1_\mathrm{diff}(\mathfrak{aff}(1|1),\mathfrak{D}_{\lambda,\mu})$.