Deformation of sl(2) and osp(1|2)-Modules of Symbols

TL;DR

This paper investigates the deformation theory of sl(2) and osp(1|2) modules on symbol spaces of differential operators, providing conditions for integrability and showing formal deformations are determined by their infinitesimal parts.

Contribution

It offers a comprehensive analysis of the deformation and integrability conditions for sl(2) and osp(1|2) modules on symbols, including the super analogue, with explicit equivalence results.

Findings

Derived necessary and sufficient conditions for deformation integrability.

Proved that all formal deformations are equivalent to their infinitesimal parts.

Extended results to the superalgebra osp(1|2).

Abstract

We consider the sl(2)-module structure on the spaces of symbols of differential opera- tors acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and we prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.