Poisson manifold


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 on a smooth manifold is a functionon the vector space of smooth functions on, making it into a Lie algebra subject to a Leibniz rule.
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, which allows one to formulate the Hamilton equations and describe the dynamics of the system through the phase space in time.
For instance, a single particle freely moving in the -dimensional Euclidean space has phase space. The coordinates describe respectively the positions and the generalised momenta. The space of observables, i.e. the smooth functions on, is naturally endowed with a binary operation called Poisson bracket, defined as. Such bracket satisfies the standard properties of a Lie bracket, plus a further compatibility with the product of functions, namely the Leibniz identity. Equivalently, the Poisson bracket on can be reformulated using the symplectic form. Indeed, if one considers the Hamiltonian vector field associated to a function, then the Poisson bracket can be rewritten as
In more abstract differential geometric terms, the configuration space is an -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 admits special coordinates where the form and the bracket are equivalent with, respectively, the symplectic form and the Poisson bracket of. Symplectic geometry is therefore the natural mathematical setting to describe classical Hamiltonian mechanics.
Poisson manifolds are further generalisations of symplectic manifolds, which arise by axiomatising the properties satisfied by the Poisson bracket on. More precisely, a Poisson manifold consists of a smooth manifold together with an abstract bracket, still called Poisson bracket, which does not necessarily arise from a symplectic form, 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 algebroids.
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. More precisely, he proved that, if two functions and are integral of motions, then there is a third function, denoted by, which is an integral of motion as well. In the Hamiltonian formulation of mechanics, where the dynamics of a physical system is described by a given function , an integral of motion is simply a function which Poisson-commutes with, i.e. such that. What will become known as Poisson's theorem can then be formulated asPoisson 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 bracket of two functions and the bracket of their associated Hamiltonian vector fields, i.e.in order to reformulate 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 correspond precisely to Lie algebra structures. Moreover, the integrability of a linear Poisson structure 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 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

Let be a smooth manifold and let denote the real algebra of smooth real-valued functions on, where the multiplication is defined pointwise. A Poisson bracket on is an -bilinear map
defining a structure of Poisson algebra on, i.e. satisfying the following three conditions:
  • Skew symmetry:.
  • Jacobi identity:.
  • Leibniz's rule:.
The first two conditions ensure that defines a Lie-algebra structure on, while the third guarantees that, for each, the linear map is a derivation of the algebra, i.e., it defines a vector field called the Hamiltonian vector field associated to.
Choosing local coordinates, any Poisson bracket is given byfor the Poisson bracket of the coordinate functions.

As bivector

A Poisson bivector on a smooth manifold is a Polyvector field satisfying the non-linear partial differential equation, where
denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing local coordinates, any Poisson bivector is given byfor skew-symmetric smooth functions on.

Equivalence of the definitions

Let be a bilinear skew-symmetric bracket satisfying Leibniz's rule; then the function can be described asfor a unique smooth bivector field. Conversely, given any smooth bivector field on, the same formula defines an almost Lie bracket 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:
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 is a sheaf of Poisson algebras. Equivalently, recall that a holomorphic bivector field on a complex manifold is a section such that. Then a holomorphic Poisson structure on is a holomorphic bivector field satisfying the equation. 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

Recall that any bivector field can be regarded as a skew homomorphism. The image consists therefore of the values of all Hamiltonian vector fields evaluated at every.
The rank of at a point is the rank of the induced linear mapping. A point is called regular for a Poisson structure on if and only if the rank of is constant on an open neighborhood of ; otherwise, it is called a singular point. Regular points form an open dense subset ; when the map is of constant rank, the Poisson structure is called regular. Examples of regular Poisson structures include trivial and nondegenerate structures.