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

TL;DR

This paper computes the first cohomology spaces of the Lie algebra rak{sl}(2) acting on spaces of n-ary differential operators on the real line, revealing the structure of these modules and their deformations.

Contribution

It provides explicit calculations of the rak{sl}(2)-cohomology for modules of n-ary differential operators, extending previous results to more complex module structures.

Findings

Explicit cohomology formulas for rak{sl}(2) modules

Classification of deformations of n-ary differential operators

Insights into module structure and representation theory

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