The spaces $\mathrm{H}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$

TL;DR

This paper computes the cohomology of a broad class of modules over the Lie superalgebra rak{osp}(1|2), with applications to differential operators on the super circle, advancing understanding of superalgebra representations.

Contribution

It provides a complete calculation of the cohomology for certain rak{osp}(1|2) modules and explores their restriction to rak{sl}(2), including applications to differential operators.

Findings

Explicit cohomology formulas for rak{osp}(1|2) modules

Analysis of restriction to rak{sl}(2) cohomology

Application to differential operators on the super circle

Abstract

We entirely compute the cohomology for a natural and large class of $\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the $\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module $M={\mathfrak D}_{\lambda,\mu}$ of differential operators on the super circle, acting on densities.