We extend to Poisson manifolds the theory of hamiltonian Lie algebroids
originally developed by two of the authors for presymplectic manifolds. As in
the presymplectic case, our definition, involving a vector bundle connection on
the Lie algebroid, reduces to the definition of hamiltonian action for an
action Lie algebroid with the trivial connection. The clean zero locus of the
momentum section of a hamiltonian Lie algebroid is an invariant coisotropic
submanifold, the distribution being given by the image of the anchor. We study
some basic examples: bundles of Lie algebras with zero anchor and cotangent and
tangent Lie algebroids. Finally, we discuss a suggestion by Alejandro Cabrera
that the conditions for a Lie algebroid A to be hamiltonian may be expressed
in terms of two bivector fields on Aβ, the natural Poisson structure on the
dual of a Lie algebroid and the horizontal lift by the connection of the given
Poisson structure on the base.Comment: 29 pages, added section on Poisson reduction, typos fixe