In this paper we study some properties of a Lie algebroid and its prolongation over the vector bundle projection of the dual bundle. We generalize some results on Poisson manifolds to the level of a Lie algebroid. The notions of canonical Poisson bivector and horizontal lift are studied and their compatibility conditions are pointed out.
Liviu Popescu. "A Note on Poisson Lie Algebroids." J. Geom. Symmetry Phys. 12 63 - 73, 2008. https://doi.org/10.7546/jgsp-12-2008-63-73