Vertical and horizontal lifts of multivector fields and applications

Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor

“…In the beginning we recall the known facts about lift of multivector fields from manifold M to T M, see [12,20,31]. Given a k-vector field in local coordinates…”

“…Section 3 contains main results of the paper. Using the concept of lifting multivectors from M to T M, see [12,20,21,31], we lift Poisson vector fields from M to Poisson vector fields on T M considered as a dual bundle to the algebroid T * M. The complete and vertical lifts of Poisson vector fields will be the building blocks for generating new Poisson structures on T M. These structures are compatible with the original algebroid structure and can be considered as infinitesimal deformations. Some of these structures belong to deformation of cohomology of Lie algebroid (the fiber-wise linear Poisson cohomology), see also [6].…”