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

Abstract

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