Deformation of Vect($\mathbb{R})$-Modules of Symbols

Abstract

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_δ^n=\bigoplus_{j=0}^n \mathcal{F}_{δ-j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space $\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{ν,μ})$ where ${\cal D}_{ν,μ}$ is the space of differential operators from $\mathcal{F}_ν$ to $\mathcal{F}_μ$. Necessary second-order integrability conditions of any infinitesimal deformations of $\mathcal{S}_δ^n$ are given. We describe completely the formal deformations for some spaces $\mathcal{S}_δ^n$ and we give concrete examples of non trivial deformations.