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 ${\mathcal{S}}_{\delta }^n={⨁}_{j=0}^n{\mathcal{F}}_{\delta -j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space ${\mathrm{H}}_{\mathrm{diff}}^2(\mathfrak{vect}(1),{\mathcal{D}}_{\nu ,\mu })$ where ${\mathcal{D}}_{\nu ,\mu }$ is the space of differential operators from ${\mathcal{F}}_{\nu }$ to ${\mathcal{F}}_{\mu }$. Necessary second-order integrability conditions of any infinitesimal deformations of ${\mathcal{S}}_{\delta }^n$ are given. We describe completely the formal deformations for some spaces ${\mathcal{S}}_{\delta }^n$ and we give concrete examples of non trivial deformations.

References in corpus (1)

• Differential operators on supercircle: conformally equivariant quantization and symbol calculus

Cited by in corpus (3)

• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols
• Deformation of sl(2) and osp(1|2)-Modules of Symbols
• Deformation of $\mathfrak{osp}(2|2)$-Modules of Symbols