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Deforming $𝔥$ -trivial the Lie algebra Vect(S¹) inside the Lie algebra of pseudodifferential operators $Ψ 𝒟 𝒪$

Imed Basdouri

Département de Mathématiques, Faculté des Sciences de Gafsa, Université de Gafsa, Gafsa, Tunisie

E-mail Address: basdourimed@yahoo.fr

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Issam Bartouli

Université Evry Val d’Essonne, Évry 91000, France

Faculté des Sciences, Département de Mathématiques, Université de Sfax, Sfax, Tunisie

E-mail Address: issambartouli@yahoo.fr

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Jean Lerbet

Université Evry Val d’Essonne, Évry 91000, France

E-mail Address: jean.lerbet@univ-evry.fr

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https://doi.org/10.1142/S0219887817500827Cited by:1 (Source: Crossref)

Abstract

In this paper, we consider the action of Vect(S¹) by Lie derivative on the spaces of pseudodifferential operators $Ψ 𝒟 𝒪$ . We study the $𝔥$ -trivial deformations of the standard embedding of the Lie algebra Vect(S¹) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle $T^{*} S^{1}$ . We classify the deformations of this action that become trivial once restricted to $𝔥$ , where $𝔥 = 𝔞 𝔣 𝔣 (1)$ or $𝔰 𝔩 (2)$ . Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

Keywords:

AMSC: 13D10, 13D03, 17B10, 14B15

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