Poisson manifold

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

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

Said in another manner, it is a Lie algebra structure on the vector space of smooth functions on ${\displaystyle M}$ such that ${\displaystyle X_{f}{\stackrel {\text{df}}{=}}\{f,\cdot \}:{C^{\infty }}(M)\to {C^{\infty }}(M)}$ is a vector field for each smooth function ${\displaystyle f}$, which we call the Hamiltonian vector field associated to ${\displaystyle f}$. 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.[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 ${\displaystyle M}$ be a smooth manifold. Let ${\displaystyle {C^{\infty }}(M)}$ denote the real algebra of smooth real-valued functions on ${\displaystyle M}$, where multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on ${\displaystyle M}$ is an ${\displaystyle \mathbb {R} }$-bilinear map

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

satisfying the following three conditions:

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

The first two conditions ensure that ${\displaystyle \{\cdot ,\cdot \}}$ defines a Lie-algebra structure on ${\displaystyle {C^{\infty }}(M)}$, while the third guarantees that for each ${\displaystyle f\in {C^{\infty }}(M)}$, the adjoint ${\displaystyle \{f,\cdot \}\colon {C^{\infty }}(M)\to {C^{\infty }}(M)}$ is a derivation of the commutative product on ${\displaystyle {C^{\infty }}(M)}$, i.e., is a vector field ${\displaystyle X_{f}}$. It follows that the bracket ${\displaystyle \{f,g\}}$ of functions ${\displaystyle f}$ and ${\displaystyle g}$ is of the form

${\displaystyle \{f,g\}=\pi (df\wedge dg)}$,

where ${\displaystyle \pi \in \Gamma {\Big (}\bigwedge ^{2}TM{\Big )}}$ is a smooth bi-vector field, called the Poisson bi-vector.

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

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

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

An integral sub-manifold for the (singular) distribution ${\displaystyle {\pi ^{\sharp }}(T^{*}M)}$ is a path-connected sub-manifold ${\displaystyle S\subseteq M}$ satisfying ${\displaystyle T_{x}S={\pi ^{\sharp }}(T_{x}^{\ast }M)}$ for all ${\displaystyle x\in S}$. Integral sub-manifolds of ${\displaystyle \pi }$ are automatically regularly immersed manifolds, and maximal integral sub-manifolds of ${\displaystyle \pi }$ are called the leaves of ${\displaystyle \pi }$. Each leaf ${\displaystyle S}$ carries a natural symplectic form ${\displaystyle \omega _{S}\in {\Omega ^{2}}(S)}$ determined by the condition ${\displaystyle [{\omega _{S}}(X_{f},X_{g})](x)=-\{f,g\}(x)}$ for all ${\displaystyle f,g\in {C^{\infty }}(M)}$ and ${\displaystyle x\in S}$. Correspondingly, one speaks of the symplectic leaves of ${\displaystyle \pi }$.[3] Moreover, both the space ${\displaystyle M_{\mathrm {reg} }}$ of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

Examples

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

Poisson maps

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

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

If ${\displaystyle \varphi \colon M\to M'}$ is also a diffeomorphism, then we call ${\displaystyle \varphi }$ an ichthyomorphism. In terms of Poisson bi-vectors, the condition that a map be Poisson is tantamount to requiring that ${\displaystyle \pi _{M}}$ and ${\displaystyle \pi _{M'}}$ be ${\displaystyle \varphi }$-related.

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

Examples of Poisson maps:

• The Cartesian product ${\displaystyle (M_{0}\times M_{1},\pi _{0}\times \pi _{1})}$ of two Poisson manifolds ${\displaystyle (M_{0},\pi _{0})}$ and ${\displaystyle (M_{1},\pi _{1})}$ is again a Poisson manifold, and the canonical projections ${\displaystyle \mathrm {pr} _{i}:M_{0}\times M_{1}\to M_{i}}$, for ${\displaystyle i\in \{0,1\}}$, 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 ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{4}}$, 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. [4][5][6]