Deformation of algebroid bracket of differential forms and Poisson manifold

TL;DR

This paper introduces new algebroid brackets on the tangent bundle of Poisson manifolds, generating novel Poisson structures and exploring their properties, including Casimir functions, especially in bi-Hamiltonian contexts.

Contribution

It constructs a family of algebroid brackets on the cotangent bundle of Poisson manifolds and derives associated Poisson structures, extending to bi-Hamiltonian cases.

Findings

New algebroid brackets on $T^*M$ for Poisson manifolds

Generation of new Poisson structures on $TM$

Methods for finding Casimir functions in these structures

Abstract

We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ 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.