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

International audienceIn 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