In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

A Poisson structure (or Poisson bracket) on a smooth manifoldon the vector space of smooth functions, making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).

Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.

Poisson geometry can be regarded as a combination of foliation theory, symplectic geometry, and Lie theory. A Poisson manifold foliates. Each leaf of the foliation has a symplectic structure. The leaves are connected transversely through Lie geometry.

Introduction

From phase spaces of classical mechanics to symplectic and Poisson manifolds

In classical mechanics, the phase space of a physical system consists of all the possible values of the position and of the momentum variables allowed by the system. It is naturally endowed with a Poisson bracket/symplectic form (see below), which allows one to formulate the Hamilton equations and describe the dynamics of the system through the phase space in time.

-dimensional Euclidean space as configuration space describe respectively the positions and the generalised momenta. The space of observables, is naturally endowed with a binary operation called Poisson bracket. Such bracket satisfies the standard properties of a Lie bracket can be reformulated using the symplectic form

-dimensional smooth manifold, and the phase space is its cotangent bundle). The latter is naturally equipped with a canonical symplectic form, which in canonical coordinates coincides with the one described above. In general, by Darboux theorem, any arbitrary symplectic manifold. Symplectic geometry is therefore the natural mathematical setting to describe classical Hamiltonian mechanics.

, but satisfies the same algebraic properties.

Poisson geometry is closely related to symplectic geometry: for instance, every Poisson bracket determines a foliation whose leaves are naturally equipped with symplectic forms. However, the study of Poisson geometry requires techniques that are usually not employed in symplectic geometry, such as the theory of Lie groupoids and algebroidss.

Moreover, there are natural examples of structures which should be "morally" symplectic, but fail to be so. For example, the smooth quotient of a symplectic manifold by a group acting by symplectomorphisms is a Poisson manifold, which in general is not symplectic. This situation models the case of a physical system which is invariant under symmetries: the "reduced" phase space, obtained by quotienting the original phase space by the symmetries, in general is no longer symplectic, but is Poisson.

History

Although the modern definition of Poisson manifold appeared only in the 1970s–1980s, its origin dates back to the nineteenth century. Alan Weinstein synthetised the early history of Poisson geometry as follows:"Poisson invented his brackets as a tool for classical dynamics. Jacobi realized the importance of these brackets and elucidated their algebraic properties, and Lie began the study of their geometry."

Indeed, Siméon Denis Poisson introduced in 1809 what we now call Poisson bracket in order to obtain new integrals of motion, i.e. quantities which are preserved throughout the motion., which is an integral of motion as well. In the Hamiltonian formulation of mechanics. What will become known as Poisson's theoremPoisson computations occupied many pages, and his results were rediscovered and simplified two decades later by Carl Gustav Jacob Jacobi. Jacobi was the first to identify the general properties of the Poisson bracket as a binary operation. Moreover, he established the relation between the (Poisson) bracket of two functions and the (Lie) bracket of their associated Hamiltonian vector fieldsin order to reformulate (and give a much shorter proof of) Poisson's theorem on integrals of motion. Jacobi's work on Poisson brackets influenced the pioneering studies of Sophus Lie on symmetries of differential equations, which led to the discovery of Lie groups and Lie algebras. For instance, what are now called linear Poisson structures (i.e. Poisson brackets on a vector space which send linear functions to linear functions) correspond precisely to Lie algebra structures. Moreover, the integrability of a linear Poisson structure (see below) is closely related to the integrability of its associated Lie algebra to a Lie group.

The twentieth century saw the development of modern differential geometry, but only in 1977 did André Lichnerowicz introduce Poisson structures as geometric objects on smooth manifolds. Poisson manifolds were further studied in the foundational 1983 paper of Alan Weinstein, where many basic structure theorems were first proved.

These works exerted a huge influence in the subsequent decades on the development of Poisson geometry, which today is a field of its own, and at the same time is deeply entangled with many others, including non-commutative geometry, integrable systems, topological field theories and representation theory.

Formal definition

There are two main points of view to define Poisson structures: it is customary and convenient to switch between them.

As bracket

, where the multiplication is defined pointwise. A Poisson bracket (or Poisson structure-bilinear map

defining a structure of Poisson algebra, i.e. satisfying the following three conditions:

• Skew symmetry.

• Jacobi identity.

• Leibniz's rule.

is a derivation, i.e., it defines a vector field called the Hamiltonian vector field.

the Poisson bracket of the coordinate functions.

As bivector

A Poisson bivector is a Polyvector field, where

denotes the Schouten–Nijenhuis bracket.

Equivalence of the definitions

that automatically obeys Leibniz's rule.

A bivector field, or the corresponding almost Lie bracket, is called an almost Poisson structure. An almost Poisson structure is Poisson if one of the following equivalent integrability conditions holds:

• satisfies the Jacobi identity (hence it is a Poisson bracket);

• (hence it a Poisson bivector);

• ;

• defines a Dirac structure which is closed under the standard Courant bracket.

Holomorphic Poisson structures

The definition of Poisson structure for real smooth manifolds can be also adapted to the complex case.

A holomorphic Poisson manifold is a complex manifold whose sheaf of holomorphic functions. Holomorphic Poisson manifolds can be characterised also in terms of Poisson–Nijenhuis structures.

Many results for real Poisson structures, e.g. regarding their integrability, extend also to holomorphic ones.

Holomorphic Poisson structures appear naturally in the context of generalised complex structures: locally, any generalised complex manifold is the product of a symplectic manifold and a holomorphic Poisson manifold.

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 distribution spanned by the Hamiltonian vector fields.

Rank of a Poisson structure

.

The rank is called regular; otherwise, it is called a singular point is called regular. Examples of regular Poisson structures include trivial and nondegenerate structures (see below).

The regular case

is a regular distribution; it is easy to check that it is involutive, therefore, by the Frobenius theorem admits a partition into leaves. Moreover, the Poisson bivector restricts nicely to each leaf, which therefore become symplectic manifolds.

The non-regular case

is singular have different dimensions.

An integral submanifold are called the leaves.

. Correspondingly, one speaks of the symplectic leaves 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).), one recovers the classical Darboux theorem for symplectic structures.

Examples

Trivial Poisson structures

carries the trivial is therefore a zero-dimensional symplectic leaf.

Nondegenerate Poisson structures

is called nondegenerate is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as symplectic manifolds.

and nondegenerate 2-forms is encoded by the musical isomorphism is closed; in such case, the bracket becomes the canonical Poisson bracketnondegenerate Poisson structures on connected itself.

Log-symplectic Poisson structures

.

, called log-symplectic.

Linear Poisson structures

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 carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (Kostant-Kirillov-Souriau are the associated structure constants.

are the orbits of the coadjoint actionand its symplectic foliation is identified with the foliation by concentric hyperboloids and conical surface (the only singular leaf being again the origin).

Fibrewise linear Poisson structures

is called fibrewise linear.

The class of vector bundles with linear Poisson structures coincides actually with that of (dual of) Lie algebroids carries a fibrewise linear Poisson bracket,.

are the connected components of the orbits of the cotangent groupoid. In general, given any algebroid orbit of the anchor map is a symplectic leaf.

.

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 a 3-vector field, which is always zero in dimension 2.

• on a 3-dimensional manifold, is automatically Poisson.

• The Cartesian product is again a Poisson manifold.

• be a (regular) foliation.

• be a Lie group acting-invariant. If the action is free and proper, the quotient manifold (namely, it is the only one such that the submersion is a Poisson map).

Poisson cohomology

The Poisson cohomology groups of a Poisson manifold are the cohomology groups of the cochain complex being Poisson.

, one obtains a morphism from the de Rham complex. 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:

• 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);

• is the space of Poisson vector fields modulo Hamiltonian vector fields;

• is the space of the infinitesimal deformations of the Poisson structure modulo trivial deformations;

• is the space of the obstructions to extend infinitesimal deformations to actual deformations.

Modular class

The modular class of a Poisson manifold is a class in the first Poisson cohomology group: for orientable manifolds, it is the obstruction to the existence of a volume form invariant under the Hamiltonian flows. It was introduced by Koszul and Weinstein.

Recall that the divergence. The modular vector field of an orientable.

, and it is called the modular class of the Poisson manifold.

An orientable Poisson manifold is called unimodular is invariant under the flow of any Hamiltonian vector field. For instance:

• Symplectic structures are always unimodular, since the Liouville form is invariant under all Hamiltonian vector fields.

• For linear Poisson structures the modular class is the infinitesimal modular character is unimodular as Poisson manifold if and only if it is unimodular as Lie algebra.

• For regular Poisson structures the modular class is related to the Reeb class of the underlying symplectic foliation (an element of the first leafwise cohomology group, which obstructs the existence of a volume normal form invariant by vector fields tangent to the foliation).

The construction of the modular class can be easily extended to non-orientable manifolds by replacing volume forms with densities.

Poisson homology

Poisson cohomology was introduced in 1977 by Lichnerowicz himself; a decade later, Brylinski introduced a homology theory.

Several results have been proved relating Poisson homology and cohomology. For instance, for orientable unimodular Poisson manifolds, Poisson homology turns out to be isomorphic to Poisson cohomology: this was proved independently by Xu and Evans-Lu-Weinstein.

Poisson maps

between Poisson manifolds is called a if it respects the Poisson structures, i.e. one of the following equivalent conditions holds (compare with the equivalent definitions of Poisson structures above):

• ;

• ;

• ;

• is a forward Dirac morphism.

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

a Poisson-diffeomorphism.

Examples

• , are Poisson maps.

• of a symplectic leaf, or of an open subset, is a Poisson map.

• between their linear Poisson structures.

• between their fibrewise linear Poisson structures.

One should notice that the notion of a Poisson map is fundamentally different from that of a symplectic map is a symplectic map, it must be an immersion.

Integration of Poisson manifolds

induces a structure of Lie algebroid, also called the cotangent algebroid.:

• 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;

• is;

• with coefficients in the trivial representation;

• .

is not always integrable to a Lie groupoid.

Symplectic groupoids

A is a Lie groupoid is asked to be a Lagrangian submanifold. The notion of symplectic groupoid was introduced at the end of the 1980s independently by several authors.

. 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, the Poisson structure is called 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. is called Poisson homotopy groupoid and is simply the Ševera-Weinstein groupoid, consisting of the quotient of the Banach space of a special class of paths can be described as an infinite-dimensional symplectic quotient.

Examples of integrations

• with the canonical symplectic structure.

• ).

• is always integrable, a symplectic groupoid being the (coadjoint.

• with the canonical symplectic form.

Symplectic realisations

A (full) symplectic realisation 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 (nondegenerate), one.

is called complete if, for any complete is complete as well. While symplectic realisations always exist for every Poisson manifold (and several different proofs are available), complete ones do not, and their existence plays a fundamental role in the integrability problem for Poisson manifolds. Indeed, using the topological obstructions to the integrability of Lie algebroids, one can show that a Poisson manifold is integrable if and only if it admits a complete symplectic realisation. This fact can also be proved more directly, without using Crainic-Fernandes obstructions.

Poisson submanifolds

A Poisson submanifold is an immersed submanifold is a Poisson map. Alternatively, one can require one of the following equivalent conditions:

• ;

• denotes the annihilator;

• .

Examples

• are Poisson submanifolds.

• , are Poisson submanifolds (actually they are unions of symplectic leaves).

• is compact, its Killing form-invariant inner product, being a union of coadjoint orbits.

Other types of submanifolds in Poisson geometry

The definition of Poisson submanifold is very natural and satisfies several good properties, e.g. the transverse intersection is not necessarily Poisson. In order to overcome this problem, one can use the notion of Poisson transversals (originally called cosymplectic submanifolds). A Poisson transversal itself), Poisson transversals are the same thing as symplectic submanifolds.

Another important generalisation of Poisson submanifolds is given by coisotropic submanifolds, introduced by Weinstein in order to "extend the lagrangian calculus from symplectic to Poisson manifolds". A coisotropic submanifold recover Poisson submanifolds, while for nondegenerate Poisson structures, coisotropic submanifolds boil down to the classical notion of coisotropic submanifold in symplectic geometry.

Other classes of submanifolds which play an important role in Poisson geometry include Lie–Dirac submanifolds, Poisson–Dirac submanifolds and pre-Poisson submanifolds.

Further topics

Deformation quantisation

The main idea of deformation quantisation is to deform the (commutative) algebra of functions on a Poisson manifold into a non-commutative one, in order to investigate the passage from classical mechanics to quantum mechanics. This topic was one of the driving forces for the development of Poisson geometry, and the precise notion of formal deformation quantisation was developed already in 1978.

A (differential) star product of formal power series.

(denoted with same symbol as the reduced Planck constant

A (formal) deformation quantisation. Several classes of Poisson manifolds have been shown to admit a canonical deformation quantisations:

• with the canonical Poisson bracket (or, more generally, any finite-dimensional vector space with a constant Poisson bracket) admits the Moyal-Weyl product;

• , with the Lie-Poisson structure, admits the Gutt star product;

• any nondegenerate Poisson manifold admits a deformation quantisation. This was shown first for symplectic manifolds with a flat symplectic connection, and then in general by de Wilde and Lecompte, while a more explicit approach was provided later by Fedosov and several other authors.

In general, building a deformation quantisation for any given Poisson manifold is a highly non trivial problem, and for several years it was not clear if it would be even possible. In 1997 Kontsevich provided a quantisation formula admits a canonical deformation quantisation; this contributed to getting him the Fields Medal in 1998.

Kontsevich's proof relies on an algebraic result, known as the formality conjecture, which involves a quasi-isomorphism of differential graded Lie algebras (with Gerstenhaber bracket and Hochschild differential). Alternative approaches and more direct constructions of Kontsevich's deformation quantisation were later provided by other authors.

Linearisation problem

is the isotropy Lie algebra is called (smoothly) linearisable.

is isomorphic to the special linear Lie algebra. Other counterexamples arise when the isotropy Lie algebra is a semisimple Lie algebra of real rank at least 2, or when it is a semisimple Lie algebra of rank 1 whose compact part (in the Cartan decomposition) is not semisimple.

A notable sufficient condition for linearisability is provided by Conn's linearisation theorem: is semisimple and compact. is semisimple but not compact. The original proof of Conn involves several estimates from analysis in order to apply the Nash-Moser theorem; a different proof, employing geometric methods which were not available at Conn's time, was provided by Crainic and Fernandes.

to be semisimple. This was conjectured by Weinstein and proved by Conn before his result in the smooth category; a more geometric proof was given by Zung. Several other particular cases when the linearisation problem has a positive answer have been proved in the formal, smooth or analytic category.

Poisson-Lie groups

A Poisson-Lie group together with a Poisson structure compatible with the multiplication map. This condition can be formulated in a number of equivalent ways:

• ;

• ;

• .

. Accordingly, a non-trivial Poisson-Lie group cannot arise from a symplectic structure, otherwise it would contradict Weinstein splitting theorem applied to cannot even be of constant rank.

induces a comultiplication with a structure of Lie coalgebra into a Lie bialgebra. Moreover, Drinfeld proved that there is an equivalence of categories between simply connected Poisson-Lie groups and finite-dimensional Lie bialgebras, extending the classical equivalence between simply connected Lie groups and finite-dimensional Lie algebras.

Weinstein generalised Poisson-Lie groups to Poisson(-Lie) groupoids., or in other equivalent ways. Moreover, Mackenzie and Xu extended Drinfeld's correspondence to a correspondence between Poisson groupoids and Lie bialgebroids.

References