Lie algebroids and Poisson-Nijenhuis structures

Poisson Structures on Lie Algebroids

In this paper the properties of Lie algebroids with Poisson structures are investigated. We generalize some results of Fernandes [1] regarding linear contravariant connections on Poisson manifolds at the level of Lie algebroids. In the last part, the notions of complete and horizontal lifts on the prolongation of Lie algebroid are studied and their compatibility conditions are pointed out.

Lie, symplectic and Poisson groupoids and their Lie algebroids

Lie algebroids associated to Poisson actions

Let P be a Poisson homogeneous G-space. In [Dr2], Drinfeld shows that corresponding to each p ∈ P , there is a maximal isotropic Lie subalgebra lp of the Lie algebra d, the double Lie algebra of the tangent Lie bialgebra (g, g∗) of G. Moreover, for g ∈ G, the two Lie algebras lp and lgp are related by lgp = Adg lp via the Adjoint action of G on d. In particular, they are isomorphic as Lie algeb...

On Poisson Realizations of Transitive Lie Algebroids

We show that every transitive Lie algebroid over a connected symplectic manifold comes from an intrinsic Lie algebroid of a symplectic leaf of a certain Poisson structure. The reconstruction of the corresponding Poisson structures from the Lie algebroid is given in terms of coupling tensors.

A Note on Poisson Lie Algebroids

The Lie algebroid [10] is a generalization of both concepts of Lie algebra and integrable distribution, being a vector bundle (E, π, M) with a Lie bracket on his space of sections with properties very similar to those of a tangent bundle. The Poisson manifolds are the smooth manifolds equipped with a Poisson bracket on their ring of functions. I have to remark that the cotangent bundle of a Poi...