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.

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].