Let be a smooth manifold. Let denote the real algebra of smooth real-valued functions on, where multiplication is defined pointwise. A Poisson bracket on is an -bilinear map satisfying the following three conditions:
Skew symmetry:.
Jacobi identity:.
Leibniz's Rule:.
The first two conditions ensure that defines a Lie-algebra structure on, while the third guarantees that for each, the adjoint is a derivation of the commutative product on, i.e., is a vector field. It follows that the bracket of functions and is of the form where is a smooth bi-vector field, called the Poisson bi-vector. Conversely, given any smooth bi-vector field on, the formula defines a bilinear skew-symmetric bracket that automatically obeys Leibniz's rule. The condition that the ensuing be a Poisson bracket — i.e., satisfy the Jacobi identity — can be characterized by the non-linear partial differential equation, where denotes the Schouten-Nijenhuis bracket on multi-vector fields. It is customary and convenient to switch between the bracket and the bi-vector points of view, and we shall do so below.
Symplectic leaves
A Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds, called its symplectic leaves. Note that a bi-vector field can be regarded as a skew homomorphism. The rank of at a point is then the rank of the induced linear mapping. Its image consists of the values of all Hamiltonian vector fields evaluated at. A point is called regular for a Poisson structure on if and only if the rank of is constant on an open neighborhood of ; otherwise, it is called a singular point. Regular points form an open dense subspace ; when, we call the Poisson structure itself regular. An integral sub-manifold for the distribution is a path-connected sub-manifold satisfying for all. Integral sub-manifolds of are automatically regularly immersed manifolds, and maximal integral sub-manifolds of are called the leaves of. Each leaf carries a natural symplectic form determined by the condition for all and. Correspondingly, one speaks of the symplectic leaves of. Moreover, both the space of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.
Examples
Every manifold carries the trivial Poisson structure.
Every symplectic manifold is Poisson, with the Poisson bi-vector equal to the inverse of the symplectic form.
The dual of a Lie algebra is a Poisson manifold. A coordinate-free description can be given as follows: naturally sits inside, and the rule for each induces a linear Poisson structure on, i.e., one for which the bracket of linear functions is again linear. Conversely, any linear Poisson structure must be of this form.
Let be a foliation of dimension on and a closed foliation two-form for which is nowhere-vanishing. This uniquely determines a regular Poisson structure on by requiring that the symplectic leaves of be the leaves of equipped with the induced symplectic form.
Poisson maps
If and are two Poisson manifolds, then a smooth mapping is called a Poisson map if it respects the Poisson structures, namely, if for all and smooth functions, we have: If is also a diffeomorphism, then we call a Poisson-diffeomorphism. In terms of Poisson bi-vectors, the condition that a map be Poisson is equivalent to requiring that and be -related. Poisson manifolds are the objects of a category, with Poisson maps as morphisms. Examples of Poisson maps:
The Cartesian product of two Poisson manifolds and is again a Poisson manifold, and the canonical projections, for, are Poisson maps.
It must be highlighted that the notion of a Poisson map is fundamentally different from that of a symplectic map. For instance, with their standard symplectic structures, there do not exist Poisson maps, whereas symplectic maps abound. One interesting, and somewhat surprising, fact is that any Poisson manifold is the codomain/image of a surjective, submersive Poisson map from a symplectic manifold.
