Poisson Structures on Cotangent Bundles

We make a study of Poisson structures of T*M which are graded structures when restricted to the fiberwise polynomial algebra, and give examples. A class of more general graded bivector fields which induce a given Poisson structure w on the base manifold M is constructed. In particular, the horizontal lifting of a Poisson structure from M to T*M via connections gives such bivector fields and we discuss the conditions for these lifts to be Poisson bivector fields and their compatibility with the canonical Poisson structure on T*M. Finally, for a 2-form on a Riemannian manifold, we study the conditions for some associated 2-forms on T*M to define Poisson structures on cotangent bundles.