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

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