Deformation of algebroid bracket of differential forms and Poisson manifold

Abstract

We construct the family of algebroid brackets $[\cdot ,\cdot {]}_{c,v}$ on the tangent bundle $T^{\ast }M$ to a Poisson manifold $(M,\pi )$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamiltonian structure $(M,{\pi }_1,{\pi }_2)$ we show how to construct another family of Poisson structures. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

31 pages