By A. Campillo

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**Example text**

This indeed satisfies the Jacobi identity for all ε, and {xi , xj }0 = π1 (x) is a linear Poisson structure. All the {·, ·}ε ’s are isomorphic for ε = 0. 4 The Cases of su(2) and sl(2; R) We shall compare the degeneracies of sl(2; R) and su(2), which are both 3-dimensional as vector spaces. First, on su(2) with coordinate functions µ1 , µ2 , µ3 , the bracket operation is defined by {µ1 , µ2 } = µ3 {µ2 , µ3 } = µ1 {µ3 , µ1 } = µ2 . The Poisson structure is trivial only at the origin. It is easy to check that the function µ21 + µ22 + µ23 is a Casimir function, meaning that it is constant along the symplectic leaves.

Qn Complete Poisson Maps Although a Poisson map ϕ : M → N preserves brackets, the image is not in general a union of symplectic leaves. Here is why: For a point x ∈ M , the image ϕ(x) lies on some symplectic leaf O in N . We can reach any other point y ∈ O from ϕ(x) by following the trajectory of (possibly more than one) hamiltonian vector field Xh . While we can lift Xh to the hamiltonian vector field Xϕ∗ h near x, knowing that Xh is complete does not ensure that Xϕ∗ h is complete. Consequently, we may not be able to lift the entire trajectory of Xh , so the point y is not necessarily in the image of ϕ.

1 (S) This is a complete Poisson map with fiber F . Exercise 21 Show that this actually defines a functor from the category of actions of π1 (S) by Poisson automorphisms on Poisson manifolds to the category of complete Poisson maps from Poisson manifolds to S. Remark. Comparing the results of this section with the theory of hamiltonian group actions, it is tempting to think of any symplectic manifold S as the “dual of the Lie algebra of π1 (S)”! 3) proved by von Neumann [127]. Chapter 9 will be devoted to analogous results in Poisson geometry.