# Poisson manifold

A **Poisson structure** on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz Rule

Said in another manner, it is a Lie-algebra structure on the vector space of smooth functions on such that is a vector field for each smooth function , which we call the **Hamiltonian vector field** associated to . These vector fields span a completely integrable singular foliation, each of whose maximal integral sub-manifolds inherits a symplectic structure. One may thus informally view a Poisson structure on a smooth manifold as a smooth partition of the ambient manifold into even-dimensional **symplectic leaves**, which are not necessarily of the same dimension.

Poisson structures are one instance of **Jacobi structures**, introduced by André Lichnerowicz in 1977.^{[1]} They were further studied in the classical paper of Alan Weinstein,^{[2]} where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few.

## Definition

Let be a smooth manifold. Let denote the real algebra of smooth real-valued functions on , where multiplication is defined pointwise. A **Poisson bracket** (or **Poisson structure**) 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.

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 (singular) 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 .^{[3]} 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 (regular) 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:

In terms of Poisson bi-vectors, the condition that a map be Poisson is tantamount 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.
- The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map.

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. ^{[4]}^{[5]}^{[6]}

## See also

## References

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