# 𝔬𝔰𝔭(1|2)-trivial deformation of 𝔬𝔰𝔭(2|2)-modules structure on the spaces of symbols 𝔖d2 of differential operators acting on the space of weighted densities 𝔉d2

## TL;DR

This paper studies how certain algebraic structures can be deformed while preserving specific properties, focusing on mathematical objects called Lie superalgebras.

## Contribution

The paper provides a classification of osp(1|2)-trivial deformations of osp(2|2)-module structures on symbol spaces.

## Key findings

- The cup-product H1∨H1 is computed for osp(2|2) relative cohomology with coefficients in differential operators.
- Integrability conditions for infinitesimal deformations of module structures are determined.
- Formal deformations are shown to be equivalent to their infinitesimal parts.

## Abstract

Let osp(2|2) be the orthosymplectic Lie superalgebra and osp(1|2) a Lie subalgebra of osp(2|2). In our paper, we describe the cup-product H1∨H1, where H1:=H1(osp(2|2),osp(1|2);Dλ,μ2) is the first differential osp(1|2)-relative cohomology of osp(2|2) with coefficients in Dλ,μ2 and Dλ,μ2:=Homdiff(Fλ2,Fμ2) is the space of linear differential operators acting on weighted densities. This result allows us to classify the osp(1|2)-trivial deformations of the osp(2|2)-module structure on the spaces of symbols Sd2. More precisely, we compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action. Furthermore, we prove that any formal osp(1|2)-trivial deformations of osp(2|2)-modules of symbols is equivalent to its infinitisemal part. This work is the simplest generalization of a result by Laraiedh [17].