# Poisson manifold

$\{fg,h\}=f\{g,h\}+g\{f,h\}$ .

Said in another manner, it is a Lie-algebra structure on the vector space of smooth functions on $M$ such that $X_{f}{\stackrel {\text{df}}{=}}\{f,\cdot \}:{C^{\infty }}(M)\to {C^{\infty }}(M)$ is a vector field for each smooth function $f$ , which we call the Hamiltonian vector field associated to $f$ . 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. They were further studied in the classical paper of Alan Weinstein, 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

$\{\cdot ,\cdot \}:{C^{\infty }}(M)\times {C^{\infty }}(M)\to {C^{\infty }}(M)$ satisfying the following three conditions:

Conversely, given any smooth bi-vector field $\pi$ on $M$ , the formula $\{f,g\}=\pi (df\wedge dg)$ defines a bilinear skew-symmetric bracket $\{\cdot ,\cdot \}$ that automatically obeys Leibniz's rule. The condition that the ensuing $\{\cdot ,\cdot \}$ be a Poisson bracket — i.e., satisfy the Jacobi identity — can be characterized by the non-linear partial differential equation $[\pi ,\pi ]=0$ , where

$[\cdot ,\cdot ]:{{\mathfrak {X}}^{p}}(M)\times {{\mathfrak {X}}^{q}}(M)\to {{\mathfrak {X}}^{p+q}}(M)$ 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.

## Poisson Maps

${\{f,g\}_{M'}}(\varphi (x))={\{f\circ \varphi ,g\circ \varphi \}_{M}}(x).$ Poisson manifolds are the objects of a category ${\mathfrak {Poiss}}$ , with Poisson maps as morphisms.

Examples of 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 $\mathbb {R} ^{2}\to \mathbb {R} ^{4}$ , 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.