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

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].

Keywords

• Cohomology
• Trivial deformation
• Lie superalgebra
• Symbol