algebra homomorphism induces a Poisson map between their linear Poisson structures. Given two Lie algebroids and , the dual of any Lie algebroid morphism over the identity induces a Poisson map between their fibrewise linear Poisson structure. One should notice that the notion of a Poisson map is fundamentally different from that of a symplectic map. For instance, with their standard symplectic structures, there exist no Poisson maps , whereas symplectic maps abound. Symplectic realisations A symplectic realisation on a Poisson manifold M consists of a symplectic manifold together with a Poisson map which is a surjective submersion. Roughly speaking, the role of a symplectic realisation is to "desingularise" a complicated (degenerate) Poisson manifold by passing to a bigger, but easier (non-degenerate), one. Notice that some authors define symplectic realisations without this last condition (so that, for instance, the inclusion of a symplectic leaf in a symplectic manifold is an example) and call full a symplectic realisation where is a surjective submersion. Examples of (full) symplectic realisations include the following: For the trivial Poisson structure , one takes as the cotangent bundle , with its canonical symplectic structure, and as the projection . For a non-degenerate Poisson structure one takes as the manifold itself and as the identity . For the Lie-Poisson structure on , one takes as the cotangent bundle of a Lie group integrating and as the dual map of the differential at the identity of the (left or right) translation . A symplectic realisation is called complete if, for any complete Hamiltonian vector field , the vector field is complete as well. While symplectic realisations always exist for every Poisson manifold (and several different proofs are available), complete ones do not, and their existence plays a fundamental role in the integrability problem for Poisson manifolds (see below). Integration of Poisson manifolds Any Poisson manifold induces a structure of Lie algebroid on its cotangent bundle , also called the cotangent algebroid. The anchor map is given by while the Lie bracket on is defined asSeveral notions defined for Poisson manifolds can be interpreted via its Lie algebroid : the symplectic foliation is the usual (singular) foliation induced by the anchor of the Lie algebroid; the symplectic leaves are the orbits of the Lie algebroid; a Poisson structure on is regular precisely when the associated Lie algebroid is; the Poisson cohomology groups coincide with the Lie algebroid cohomology groups