Poisson manifold

A Poisson structure on a smooth manifold $M$ is a Lie bracket $\{ \cdot,\cdot \}$ (called a Poisson bracket in this special case) on the algebra ${C^{\infty}}(M)$ of smooth functions on $M$, subject to the Leibniz Rule

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

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

Let $M$ be a smooth manifold. Let ${C^{\infty}}(M)$ denote the real algebra of smooth real-valued functions on $M$, where multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on $M$ is an $\mathbb{R}$-bilinear map

$\{ \cdot,\cdot \}: {C^{\infty}}(M) \times {C^{\infty}}(M) \to {C^{\infty}}(M)$

satisfying the following three conditions:

• Skew symmetry: $\{ f,g \} = - \{ g,f \}$.
• Jacobi identity: $\{ f,\{ g,h \} \} + \{ g,\{ h,f \} \} + \{ h,\{ f,g \} \} = 0$.
• Leibniz's Rule: $\{ f g,h \} = f \{ g,h \} + g \{ f,h \}$.

The first two conditions ensure that $\{ \cdot,\cdot \}$ defines a Lie-algebra structure on ${C^{\infty}}(M)$, while the third guarantees that for each $f \in {C^{\infty}}(M)$, the adjoint $\{ f,\cdot \}: {C^{\infty}}(M) \to {C^{\infty}}(M)$ is a derivation of the commutative product on ${C^{\infty}}(M)$, i.e., is a vector field $X_{f}$. It follows that the bracket $\{ f,g \}$ of functions $f$ and $g$ is of the form $\{ f,g \} = \pi(df \wedge dg)$, where $\pi \in \Gamma \left( \bigwedge^{2} T M \right)$ is a smooth bi-vector field.

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

$[\cdot,\cdot]: {\mathfrak{X}^{p}}(M) \times {\mathfrak{X}^{q}}(M) \to {\mathfrak{X}^{p + q - 1}}(M)$

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.

Note that a bi-vector field can be regarded as a skew homomorphism $\pi^{\sharp}: T^{*} M \to T M$. The rank of $\pi$ at a point $x \in M$ is then the rank of the induced linear mapping $\pi^{\sharp}_{x}$. Its image consists of the values ${X_{f}}(x)$ of all Hamiltonian vector fields evaluated at $x$. A point $x \in M$ is called regular for a Poisson structure $\pi$ on $M$ if and only if the rank of $\pi$ is constant on an open neighborhood of $x \in M$; otherwise, it is called a singular point. Regular points form an open dense subspace $M_{\mathrm{reg}} \subseteq M$; when $M_{\mathrm{reg}} = M$, we call the Poisson structure itself regular.

An integral sub-manifold for the (singular) distribution ${\pi^{\sharp}}(T^{*} M)$ is a path-connected sub-manifold $S \subseteq M$ satisfying $T_{x} S = {\pi^{\sharp}}(T^{\ast}_{x} M)$ for all $x \in S$. Integral sub-manifolds of $\pi$ are automatically regularly immersed manifolds, and maximal integral sub-manifolds of $\pi$ are called the leaves of $\pi$. Each leaf $S$ carries a natural symplectic form $\omega_{S} \in {\Omega^{2}}(S)$ determined by the condition $[{\omega_{S}}(X_{f},X_{g})](x) = - \{ f,g \}(x)$ for all $f,g \in {C^{\infty}}(M)$ and $x \in S$. Correspondingly, one speaks of the symplectic leaves of $\pi$. Moreover, both the space $M_{\mathrm{reg}}$ of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

Examples

• Every manifold $M$ carries the trivial Poisson structure $\{ f,g \} = 0$.
• Every symplectic manifold $(M,\omega)$ is Poisson, with the Poisson bi-vector $\pi$ equal to the inverse $\omega^{-1}$ of the symplectic form $\omega$.
• The dual $\mathfrak{g}^{*}$ of a Lie algebra $(\mathfrak{g},[\cdot,\cdot])$ is a Poisson manifold. A coordinate-free description can be given as follows: $\mathfrak{g}$ naturally sits inside ${C^{\infty}}(\mathfrak{g}^{*})$, and the rule $\{ X,Y \} \stackrel{\text{df}}{=} [X,Y]$ for each $X,Y \in \mathfrak{g}$ induces a linear Poisson structure on $\mathfrak{g}^{*}$, i.e., one for which the bracket of linear functions is again linear. Conversely, any linear Poisson structure must be of this form.
• Let $\mathcal{F}$ be a (regular) foliation of dimension $2 r$ on $M$ and $\omega \in {\Omega^{2}}(\mathcal{F})$ a closed foliation two-form for which $\omega^{r}$ is nowhere-vanishing. This uniquely determines a regular Poisson structure on $M$ by requiring that the symplectic leaves of $\pi$ be the leaves $S$ of $\mathcal{F}$ equipped with the induced symplectic form $\omega|_S$.

Poisson Maps

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, namely, if for all $x \in M$ and smooth functions $f,g \in {C^{\infty}}(M')$, we have:

${\{ f,g \}_{M'}}(\varphi(x)) = {\{ f \circ \varphi,g \circ \varphi \}_{M}}(x).$

In terms of Poisson bi-vectors, the condition that a map be Poisson is tantamount to requiring that $\pi_{M}$ and $\pi_{M'}$ be $\varphi$-related.

Poisson manifolds are the objects of a category $\mathfrak{Poiss}$, with Poisson maps as morphisms.

Examples of Poisson maps:

• The Cartesian product $(M_{0} \times M_{1},\pi_{0} \times \pi_{1})$ of two Poisson manifolds $(M_{0},\pi_{0})$ and $(M_{1},\pi_{1})$ is again a Poisson manifold, and the canonical projections $\mathrm{pr}_{i}: M_{0} \times M_{1} \to M_{i}$, for $i \in \{ 0,1 \}$, 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 $\mathbb{R}^{2} \to \mathbb{R}^{4}$, 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.

See also

• Nambu-Poisson manifold
• Poisson–Lie group
• Poisson supermanifold

References

• [2]
• Available at thesis
• See also the review by Ping Xu in the Bulletin of the AMS.