Poisson manifold

In differential geometry, a Poisson structure on a smooth manifold $M$ is a Lie bracket $\$ (called a Poisson bracket in this special case) on the algebra $(M)$ of smooth functions on $M$, subject to the Leibniz rule

:$\ = \h + g \$.

Equivalently, $\$ defines a Lie algebra structure on the vector space $(M)$ of smooth functions on $M$ such that $X_:= \: (M) \to (M)$ is a vector field for each smooth function $f$ (making $(M)$ into a Poisson algebra).

Poisson structures on manifolds were 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.

Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.

Definition

There are two main points of view to define Poisson structures: it is customary and convenient to switch between them, and we shall do so below.

As bracket

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

:$\: (M) \times (M) \to (M)$

defining a structure of Poisson algebra on $(M)$, i.e. satisfying the following three conditions:
* Skew symmetry: $\ = - \$.
* Jacobi identity: $\ + \ + \ = 0$.
* Leibniz's Rule: $\ = f \ + g \$.

The first two conditions ensure that $\$ defines a Lie-algebra structure on $(M)$, while the third guarantees that, for each $f \in (M)$, the linear map $X_f := \: (M) \to (M)$ is a derivation of the algebra $(M)$, i.e., it defines a vector field $X_ \in \mathfrak(M)$ called the Hamiltonian vector field associated to $f$.

Choosing local coordinates $(U, x^i)$, any Poisson bracket is given by$\_ = \sum_ \pi^ \frac \frac,$for $\pi^ = \$ the Poisson bracket of the coordinate functions.

As bivector

A Poisson bivector on a smooth manifold $M$ is a bivector field $\pi \in \mathfrak^2(M) := \Gamma \big( \wedge^ T M \big)$ satisfying the non-linear partial differential equation pi,\pi= 0 , where

: cdot,\cdot (M) \times (M) \to (M)

denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing local coordinates $(U, x^i)$, any Poisson bivector is given by$\pi_ = \sum_ \pi^ \frac \frac,$for $\pi^$ skew-symmetric smooth functions on $U$.

Equivalence of the definitions

Let $\$ be a bilinear skew-symmetric bracket satisfying Leibniz's rule; then the function $\$ can be described as

:$\ = \pi(df \wedge dg)$,

for a unique smooth bivector field $\pi \in \mathfrak^2(M)$. Conversely, given any smooth bivector field $\pi$ on $M$, the same formula $\ = \pi(df \wedge dg)$ defines a bilinear skew-symmetric bracket $\$ that automatically obeys Leibniz's rule.

Last, the following conditions are equivalent

* $\$ satisfies the Jacobi identity (hence it is a Poisson bracket)
* $\pi$ satisfies pi,\pi= 0 (hence it a Poisson bivector)
* the map $(M) \to \mathfrak(M), f \mapsto X_f$ is a Lie algebra homomorphism, i.e. the Hamiltonian vector fields satisfy _f, X_g= X_
* the graph $Graph(\pi) \subset TM \oplus T^*M$ defines a Dirac structure, i.e. a Lagrangian subbundle $D \subset TM \oplus T^*M$ which is closed under the standard Courant bracket.

Symplectic leaves

A Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds of possibly different dimensions, called its symplectic leaves. These arise as the maximal integral submanifolds of the completely integrable singular foliation spanned by the Hamiltonian vector fields.

Rank of a Poisson structure

Recall that any bivector field can be regarded as a skew homomorphism $\pi^: T^ M \to T M, \alpha \mapsto \pi(\alpha,\cdot)$. The image $(T^ M) \subset TM$ consists therefore of the values $(x)$ of all Hamiltonian vector fields evaluated at every $x \in M$.

The rank of $\pi$ at a point $x \in M$ is the rank of the induced linear mapping $\pi^_$. 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 $M_ \subseteq M$; when $M_ = M$, i.e. the map $\pi^\sharp$ is of constant rank, the Poisson structure $\pi$ is called regular. Examples of regular Poisson structures include trivial and nondegenerate structures (see below).

The regular case

For a regular Poisson manifold, the image $(T^ M) \subset TM$ is a regular distribution; it is easy to check that it is involutive, therefore, by Frobenius theorem, $M$ admits a partition into leaves. Moreover, the Poisson bivector restricts nicely to each leaf, which become therefore symplectic manifolds.

The non-regular case

For a non-regular Poisson manifold the situation is more complicated, since the distribution $(T^ M) \subset TM$ is singular, i.e. the vector subspaces $(T^_x M) \subset T_xM$ have different dimensions.

An integral submanifold for $(T^ M)$ is a path-connected submanifold $S \subseteq M$ satisfying $T_ S = (T^_ M)$ for all $x \in S$. Integral submanifolds of $\pi$ are automatically regularly immersed manifolds, and maximal integral submanifolds of $\pi$ are called the leaves of $\pi$.

Moreover, each leaf $S$ carries a natural symplectic form $\omega_ \in (S)$ determined by the condition X_,X_)x) = - \(x) for all $f,g \in (M)$ and $x \in S$. Correspondingly, one speaks of the symplectic leaves of $\pi$. Moreover, both the space $M_$ of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

Weinstein splitting theorem

To show the existence of symplectic leaves also in the non-regular case, one can use Weinstein splitting theorem (or Darboux-Weinstein theorem). It states that any Poisson manifold $(M^n, \pi)$ splits locally around a point $x_0 \in M$ as the product of a symplectic manifold $(S^, \omega)$ and a transverse Poisson submanifold $(T^, \pi_T)$ vanishing at $x_0$. More precisely, if $\mathrm(\pi_) = 2k$, there are local coordinates $(U, p_1,\ldots,p_k,q^1,\ldots, q^k,x^1,\ldots,x^)$ such that the Poisson bivector $\pi$ splits as the sum$\pi_ = \sum_^ \frac \frac + \frac \sum_^ \phi^(x) \frac \frac,$where $\phi^(x_0) = 0$. Note that, when the rank of $\pi$ is maximal (e.g. the Poisson structure is nondegenerate), one recovers the classical Darboux theorem for symplectic structures.

Examples

Trivial Poisson structures

Every manifold $M$ carries the trivial Poisson structure $\ = 0$, equivalently described by the bivector $\pi=0$. Every point of $M$ is therefore a zero-dimensional symplectic leaf.

Nondegenerate Poisson structures

A bivector field $\pi$ is called nondegenerate if $\pi^: T^ M \to T M$ is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as symplectic manifolds $(M,\omega)$.

Indeed, there is a bijective correspondence between nondegenerate bivector fields $\pi$ and nondegenerate 2-forms $\omega$, given by$\pi^\sharp = (\omega^)^,$where $\omega$ is encoded by $\omega^: TM \to T^*M, \quad v \mapsto \omega(v,\cdot)$. Furthermore, $\pi$ is Poisson precisely if and only if $\omega$ is closed; in such case, the bracket becomes the canonical Poisson bracket from Hamiltonian mechanics:$\ := \omega (X_f,X_g).$Non-degenerate Poisson structures have only one symplectic leaf, namely $M$ itself, and their Poisson algebra $(\mathcal^(M), \)$ become a Poisson ring.

Linear Poisson structures

A Poisson structure $\$ on a vector space $V$ is called linear when the bracket of two linear functions is still linear.

The class of vector spaces with linear Poisson structures coincides actually with that of (dual of) Lie algebras. Indeed, the dual $\mathfrak^$ of any finite-dimensional Lie algebra (\mathfrak, cdot,\cdot carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS ( Kostant- Kirillov- Souriau) structure: \ (\xi) := \xi ( _\xi f,d_\xi g), where $f,g \in \mathcal^(\mathfrak^*), \xi \in \mathfrak^*$ and the derivatives $d_\xi f, d_\xi g: T_ \mathfrak^* \to \mathbb^n$ are interpreted as elements of the bidual $\mathfrak^ \cong \mathfrak$. Equivalently, the Poisson bivector can be locally expressed as$\pi = \sum_ c^_k x^k \frac \frac,$where $x^i$ are coordinates on $\mathfrak^$ and $c_k^$ are the associated structure constants of $\mathfrak$,

Conversely, any linear Poisson structure $\$ on $V$ must be of this form, i.e. there exists a natural Lie algebra structure induced on $\mathfrak:=V^*$ whose Lie-Poisson bracket recovers $\$.

The symplectic leaves of the Lie-Poisson structure on $\mathfrak^*$ are the orbits of the coadjoint action of $G$ on $\mathfrak^*$.

Fibrewise linear Poisson structures

The previous example can be generalised as follows. A Poisson structure on the total space of a vector bundle $E \to M$ is called fibrewise linear when the bracket of two smooth functions $E \to \mathbb$, whose restrictions to the fibres are linear, is still linear when restricted to the fibres. Equivalently, the Poisson bivector field $\pi$ is asked to satisfy $(m_t)^*\pi = t \pi$ for any $t >0$, where $m_t: E \to E$ is the scalar multiplication $v \mapsto tv$.

The class of vector bundles with linear Poisson structures coincides actually with that of (dual of) Lie algebroids. Indeed, the dual $A^*$ of any Lie algebroid (A, cdot, \cdot carries a fibrewise linear Poisson bracket, uniquely defined by$\:= ev_ \quad \quad \forall \alpha, \beta \in \Gamma(A),$where $\mathrm_\alpha: A^* \to \mathbb, \phi \mapsto \phi(\alpha)$ is the evaluation by $\alpha$. Equivalently, the Poisson bivector can be locally expressed as$\pi = \sum_ B^i_a(x) \frac \frac + \sum_ C_^c(x) y_c \frac \frac,$where $x^i$ are coordinates around a point $x \in M$, $y_a$ are fibre coordinates on $A^*$, dual to a local frame $e_a$ of $A$, and $B^i_a$ and $C^c_$ are the structure function of $A$, i.e. the unique smooth functions satisfying \rho(e_a) = \sum_i B^i_a (x) \frac, \quad \quad _a, e_b= \sum_c C^c_ (x) e_c. Conversely, any fibrewise linear Poisson structure $\$ on $E$ must be of this form, i.e. there exists a natural Lie algebroid structure induced on $A:=E^*$ whose Lie-Poisson backet recovers $\$.

The symplectic leaves of $A^*$ are the cotangent bundles of the algebroid orbits $\mathcal \subseteq A$; equivalently, if $A$ is integrable to a Lie groupoid $\mathcal \rightrightarrows M$, they are the connecting components of the orbits of the cotangent groupoid $T^* \mathcal \rightrightarrows A^*$.

For $M = \$ one recovers linear Poisson structures, while for $A = TM$ the fibrewise linear Poisson structure is the nondegenerate one given by the canonical symplectic structure of $T^*M$.

Other examples and constructions

* Any constant bivector field on a vector space is automatically a Poisson structure; indeed, all three terms in the Jacobiator are zero, being the bracket with a constant function.
*Any bivector field on a 2-dimensional manifold is automatically a Poisson structure; indeed, pi,\pi is a 3-vector field, which is always zero in dimension 2.
*Given any Poisson bivector field $\pi$ on a 3-dimensional manifold $M$, the bivector field $f \pi$, for any $f \in \mathcal^\infty(M)$, is automatically Poisson.
*The Cartesian product $(M_ \times M_,\pi_ \times \pi_)$ of two Poisson manifolds $(M_,\pi_)$ and $(M_,\pi_)$ is again a Poisson manifold.
*Let $\mathcal$ be a (regular) foliation of dimension $2 r$ on $M$ and $\omega \in (\mathcal)$ a closed foliation two-form for which the power $\omega^$ is nowhere-vanishing. This uniquely determines a regular Poisson structure on $M$ by requiring the symplectic leaves of $\pi$ to be the leaves $S$ of $\mathcal$ equipped with the induced symplectic form $\omega, _S$.
*Let $G$ be a Lie group acting on a Poisson manifold $(M,\pi)$ by Poisson diffeomorphisms. If the action is free and proper, the quotient manifold $M/G$ inherits a Poisson structure $\pi_$ from $\pi$ (namely, it is the only one such that the submersion $(M,\pi) \to (M/G,\pi_)$ is a Poisson map).

Poisson cohomology

The Poisson cohomology groups $H^k(M,\pi)$ of a Poisson manifold are the cohomology groups of the cochain complex$\ldots \xrightarrow \mathfrak^\bullet(M) \xrightarrow \mathfrak^(M) \xrightarrow \ldots \color$

where d_\pi = pi,- is the Schouten-Nijenhuis bracket with $\pi$. Note that such a sequence can be defined for every bivector on $M$; the condition $d_\pi \circ d_\pi = 0$ is equivalent to pi,\pi0 , i.e. $M$ being Poisson.

Using the morphism $\pi^: T^ M \to T M$, one obtains a morphism from the de Rham complex $(\Omega^\bullet(M),d_)$ to the Poisson complex $(\mathfrak^\bullet(M), d_\pi)$, inducing a group homomorphism $H_^\bullet(M) \to H^\bullet(M,\pi)$. In the nondegenerate case, this becomes an isomorphism, so that the Poisson cohomology of a symplectic manifold fully recovers its de Rham cohomology.

Poisson cohomology is difficult to compute in general, but the low degree groups contain important geometric information on the Poisson structure:

* $H^0(M,\pi)$ is the space of the Casimir functions, i.e. smooth functions Poisson-commuting with all others (or, equivalently, smooth functions constant on the symplectic leaves)
*$H^1(M,\pi)$ is the space of Poisson vector fields modulo hamiltonian vector fields
* $H^2(M,\pi)$ is the space of the infinitesimal deformations of the Poisson structure modulo trivial deformations
* $H^3(M,\pi)$ is the space of the obstructions to extend infinitesimal deformations to actual deformations.

Poisson maps

A smooth map $\varphi: M \to N$ between Poisson manifolds is called a if it respects the Poisson structures, i.e. one of the following equivalent conditions holds (see the various definitions of Poisson structures above):

* the Poisson brackets $\_$ and $\_$ satisfy $(\varphi(x)) = (x)$ for every $x \in M$ and smooth functions $f,g \in (N)$
* the bivector fields $\pi_$ and $\pi_$ are $\varphi$-related, i.e. $\pi_N = \varphi_* \pi_M$

* the Hamiltonian vector fields associated to every smooth function $H \in \mathcal^\infty(N)$ are $\varphi$-related, i.e. $X_H = \varphi_* X_$
* the differential $d\varphi: (TM,Graph(\pi_M)) \to (TN,Graph(\pi_N))$ is a Dirac morphism.

An anti-Poisson map satisfies analogous conditions with a minus sign on one side.

Poisson manifolds are the objects of a category $\mathfrak$, with Poisson maps as morphisms. If a Poisson map $\varphi: M\to N$ is also a diffeomorphism, then we call $\varphi$ a Poisson-diffeomorphism.

Examples

* Given the product Poisson manifold $(M_ \times M_,\pi_ \times \pi_)$, the canonical projections $\mathrm_: M_ \times M_ \to M_$, for $i \in \$, are Poisson maps.
* The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map.
*Given two Lie algebras $\mathfrak$ and $\mathfrak$, the dual of any Lie algebra homomorphism $\mathfrak \to \mathfrak$ induces a Poisson map $\mathfrak^* \to \mathfrak^*$ between their linear Poisson structures.
*Given two Lie algebroids $A \to M$ and $B \to M$, the dual of any Lie algebroid morphism $A \to B$ over the identity induces a Poisson map $B^* \to A^*$ between their fibrewise linear Poisson structure.

One should note that the notion of a Poisson map is fundamentally different from that of a symplectic map. For instance, with their standard symplectic structures, there exist no Poisson maps $\mathbb^ \to \mathbb^$, whereas symplectic maps abound.

Symplectic realisations

A symplectic realisation on a Poisson manifold M consists of a symplectic manifold $(P,\omega)$ together with a Poisson map $\phi: (P,\omega) \to (M,\pi)$ which is a surjective submersion. Roughly speaking, the role of a symplectic realisation is to "desingularise" a complicated (degenerate) Poisson manifold by passing to a bigger, but easier (non-degenerate), one.

Note that some authors define symplectic realisations without this last condition (so that, for instance, the inclusion of a symplectic leaf in a symplectic manifold is an example) and call full a symplectic realisation where $\phi$ is a surjective submersion. Examples of (full) symplectic realisations include the following:

* For the trivial Poisson structure $(M,0 )$, one takes the cotangent bundle $T^*M$, with its canonical symplectic structure, and the projection $T^*M \to M$.
* For a non-degenerate Poisson structure $(M,\omega)$ one takes $M$ itself and the identity $M \to M$.

* For the Lie-Poisson structure on $\mathfrak^*$, one takes the cotangent bundle $T^*G$ of a Lie group $G$ integrating $\mathfrak$ and the dual map $\phi: T^*G \to \mathfrak^*$ of the differential at the identity of the (left or right) translation $G \to G$.
A symplectic realisation $\phi$ is called complete if, for any complete Hamiltonian vector field $X_H$, the vector field $X_$ is complete as well. While symplectic realisations always exist for every Poisson manifold (several different proofs are available), complete ones play a fundamental role in the integrability problem for Poisson manifolds (see below).

Integration of Poisson manifolds

Any Poisson manifold $(M,\pi)$ induces a structure of Lie algebroid on its cotangent bundle $T^*M \to M$, also called the cotangent algebroid. The anchor map is given by $\pi^: T^ M \to T M$ while the Lie bracket on $\Gamma(T^*M) = \Omega^1(M)$ is defined as alpha, \beta:= \mathcal_ (\beta) - \iota_ d\alpha = \mathcal_ (\beta) - \mathcal_ (\alpha) - d\pi (\alpha, \beta). Several notions defined for Poisson manifolds can be interpreted via its Lie algebroid $T^*M$:

* the symplectic foliation is the usual (singular) foliation induced by the anchor of the Lie algebroid
*the symplectic leaves are the orbits of the Lie algebroid
* a Poisson structure on $M$ is regular precisely when the associated Lie algebroid $T^*M$ is
* the Poisson cohomology groups coincide with the Lie algebroid cohomology groups of $T^*M$ with coefficients in the trivial representation.

It is of crucial importance to notice that the Lie algebroid $T^*M$ is not always integrable to a Lie groupoid.

Symplectic groupoids

A is a Lie groupoid $\mathcal \rightrightarrows M$ together with a symplectic form $\omega \in \Omega^2(\mathcal)$ which is also multiplicative (i.e. compatible with the groupoid structure). Equivalently, the graph of $\omega$ is asked to be a Lagrangian submanifold of $(\mathcal \times \mathcal \times \mathcal, \omega \oplus \omega \oplus - \omega)$. Among the several consequences, the dimension of $\mathcal$ is automatically twice the dimension of $M$.

A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure $\pi$ such that the source map $s: (\mathcal, \omega) \to (M,\pi)$ and the target map $t: (\mathcal, \omega) \to (M,\pi)$ are, respectively, a Poisson map and an anti-Poisson map. Moreover, the Lie algebroid $Lie(\mathcal)$ is isomorphic to the cotangent algebroid $T^*M$ associated to the Poisson manifold $(M,\pi)$. Conversely, if the cotangent bundle $T^*M$ of a Poisson manifold is integrable to some Lie groupoid $\mathcal \rightrightarrows M$, then $\mathcal$ is automatically a symplectic groupoid.

Accordingly, the integrability problem for a Poisson manifold consists in finding a (symplectic) Lie groupoid which integrates its cotangent algebroid; when this happens, we say that the Poisson structure is integrable.

While any Poisson manifold admits a local integration (i.e. a symplectic groupoid where the multiplication is defined only locally), there are general topological obstructions to its integrability, coming from the integrability theory for Lie algebroids. Using such obstructions, one can show that a Poisson manifold is integrable if and only if it admits a complete symplectic realisation.

The candidate $\Pi(M,\pi)$ for the symplectic groupoid integrating a given Poisson manifold $(M,\pi)$ is called Poisson homotopy groupoid and is simply the Weinstein groupoid of the cotangent algebroid $T^*M \to M$, consisting of the quotient of the Banach space of a special class of paths in $T^*M$ up to a suitable equivalent relation. Equivalently, $\Pi(M,\pi)$ can be described as an infinite-dimensional symplectic quotient.

Examples of integrations

* The trivial Poisson structure $(M,0)$ is always integrable, the symplectic groupoid being the bundle of abelian (additive) groups $T^*M \rightrightarrows M$ with the canonical symplectic form.
* A non-degenerate Poisson structure on $M$ is always integrable, the symplectic groupoid being the pair groupoid $M \times M \rightrightarrows M$ together with the symplectic form $s^* \omega - t^* \omega$ (for$\pi^\sharp = (\omega^)^$).
* A Lie-Poisson structure on $\mathfrak^*$ is always integrable, the symplectic groupoid being the ( coadjoint) action groupoid $G \times \mathfrak^* \rightrightarrows \mathfrak^*$, for $G$ the simply connected integration of $\mathfrak$, together with the canonical symplectic form of $T^*G \cong G \times \mathfrak^*$.
* A Lie-Poisson structure on $A^*$ is integrable if and only if the Lie algebroid $A \to M$ is integrable to a Lie groupoid $\mathcal \rightrightarrows M$, the symplectic groupoid being the cotangent groupoid $T^*\mathcal \rightrightarrows A^*$ with the canonical symplectic form.

Submanifolds

A Poisson submanifold of $(M, \pi)$ is an immersed submanifold $N \subseteq M$ such that the immersion map $(N,\pi_) \hookrightarrow (M,\pi)$ is a Poisson map. Equivalently, one asks that every Hamiltonian vector field $X_f$, for $f \in \mathcal^\infty(M)$, is tangent to $N$.

This definition is very natural and satisfies several good properties, e.g. the transverse intersection of two Poisson submanifolds is again a Poisson submanifold. However, it has also a few problems:

* Poisson submanifolds are rare: for instance, the only Poisson submanifolds of a symplectic manifold are the open sets;
* the definition does not behave functorially: if $\Phi: (M,\pi_M) \to (N,\pi_N)$ is a Poisson map transverse to a Poisson submanifold $Q$ of $N$, the submanifold $\Phi^ (Q)$ of $M$ is not necessarily Poisson.

In order to overcome these problems, one often uses the notion of a Poisson transversal (originally called cosymplectic submanifold). This can be defined as a submanifold $X \subseteq M$ which is transverse to every symplectic leaf $S$ and such that the intersection $X \cap S$ is a symplectic submanifold of $(S,\omega_S)$. It follows that any Poisson transversal $X \subseteq (M,\pi)$ inherits a canonical Poisson structure $\pi_X$ from $\pi$. In the case of a nondegenerate Poisson manifold $(M, \pi)$ (whose only symplectic leaf is $M$ itself), Poisson transversals are the same thing as symplectic submanifolds.

More general classes of submanifolds play an important role in Poisson geometry, including Lie-Dirac submanifolds, Poisson-Dirac submanifolds, coisotropic submanifolds and pre-Poisson submanifolds.

See also

* Nambu-Poisson manifold
* Poisson–Lie group
* Poisson supermanifold
*Kontsevich quantization formula

References

Books and surveys

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{{Manifolds

Differential geometry
Symplectic geometry
Smooth manifolds
Structures on manifolds