Deforming $h$-trivial the Lie algebra Vect($S^1$) inside the Lie algebra of pseudodifferential operators $Ψ\mathcal{DO}$

Imed Basdouri 1 Issam Bartouli 2, 3 Jean Lerbet 2
1 Département de Mathématiques, Faculté des Sciences de Gafsa
2 IBISC - Informatique, Biologie Intégrative et Systèmes Complexes
3 Faculté des Sciences, Département de Mathématiques
Abstract : In this paper, we consider the action of Vect($S^1$) by Lie derivative on the spaces of pseudodifferential operators $Ψ\mathcal{DO}$. 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 $h=aff(1)$ or $si(2)$. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.
Keywords : Lie algebras Deformation theory cohomology Poisson Lie algebra
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Imed Basdouri, Issam Bartouli, Jean Lerbet. Deforming $h$-trivial the Lie algebra Vect($S^1$) inside the Lie algebra of pseudodifferential operators $Ψ\mathcal{DO}$. International Journal of Geometric Methods in Modern Physics, World Scientific Publishing, 2017, 14 (6), ⟨10.1142/S0219887817500827⟩. ⟨hal-01528767⟩

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