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

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

2007-02-22

arxiv.org/abs/math/0702664v2

Journal of Geometry and Physics 60 (2010) 531-542

Mathematics

Representation Theory

Scientific paper

10.1016/j.geomphys.2009.12.002

We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}$. We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space $\mathrm{H}^2_{\rm diff}(\mathfrak{vect}(1),{\cal D}_{\nu,\mu})$ where ${\cal D}_{\nu,\mu}$ 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.

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