Poisson structures on tangent bundles

✍ Scribed by Gabriel Mitric; Izu Vaisman

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 188 KB
• Volume: 18
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(02)00148-1

✦ Synopsis

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle T M by means of the Schouten-Nijenhuis bracket of covariant symmetric tensor fields defined by the cotangent Lie algebroid of M. Then, we discuss Poisson structures of T M which have a graded restriction to the fiberwise polynomial algebra; they must be π-related (π : T M → M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of T M which produce graded brackets on the fiberwise polynomial algebra, and are transversal Poisson structures of the foliation by fibers. Finally, such bivector fields are produced by a process of horizontal lifting of Poisson structures from M to T M via connections.