The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM 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 TM 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 TM via connections.
Bibliographical noteFunding Information:
During the work on this paper, Gabriel Mitric held a postdoctoral grant at the University of Haifa, Israel. He wants to thank the University of Haifa, its Department of Mathematics, and, personally, Professor Izu Vaisman for the grant and for hospitality during the period when the postdoctoral program was completed.
- Graded bivector field
- Poisson structure
- Tangent bundle
ASJC Scopus subject areas
- Geometry and Topology
- Computational Theory and Mathematics