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

TL;DR

This paper demonstrates that complete and vertical lifts of Poisson vector fields preserve the Poisson structure on tangent bundles and explores the resulting infinitesimal deformations of the Poisson tensor.

Contribution

It establishes that lifts of Poisson vector fields remain Poisson and characterizes the infinitesimal deformations of the lifted Poisson tensor.

Findings

Lifts of Poisson vector fields are Poisson on tangent bundles.

Infinitesimal deformations of the Poisson tensor are described.

Examples in low-dimensional cases illustrate the theory.

Abstract

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.