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In this paper we prove that both complete and vertical lifts of a Poissonvector field from a Poisson manifold $(M, \pi)$ to its tangent bundle $(TM,\pi_{TM})$ are also Poisson. We use this fact to describe the infinitesimaldeformations of Poisson tensor $\pi_{TM}$. We study some of their propertiesand present a extensive set of examples in a low dimensional case.

Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor