In
differential geometry, a Poisson structure on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is a
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
(called a
Poisson bracket in this special case) on the algebra
of
smooth functions
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
on
, subject to the
Leibniz rule
:
.
Equivalently,
defines a
Lie algebra structure on the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of
smooth functions on
such that
is a
vector field for each smooth function
(making
into a
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
).
Poisson structures on manifolds were introduced by
André Lichnerowicz
André Lichnerowicz (January 21, 1915, Bourbon-l'Archambault – December 11, 1998, Paris) was a noted French differential geometer and mathematical physicist of Polish descent. He is considered the founder of modern Poisson geometry.
Biograp ...
in 1977.
[ ] They were further studied in the classical paper of
Alan Weinstein
Alan David Weinstein (17 June, 1943, New York City) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson geometry.
Education and career
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
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
,
topological field theories and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, to name a few.
Poisson structures are named after the French mathematician
Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
, 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
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 (or Poisson structure) on
is an
-
bilinear map
:
defining a structure of
Poisson algebra In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central ...
on
, i.e. satisfying the following three conditions:
*
Skew symmetry
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if ...
:
.
*
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
:
.
*
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
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
of the algebra
, i.e., it defines a
vector field called the
Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ...
associated to
.
Choosing local coordinates
, any Poisson bracket is given by
for
the Poisson bracket of the coordinate functions.
As bivector
A Poisson bivector on a smooth manifold
is a
bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
field
satisfying the non-linear partial differential equation
, where
:
denotes the
Schouten–Nijenhuis bracket In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two differ ...
on multivector fields. Choosing local coordinates
, any Poisson bivector is given by
for
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 as
:
,
for a unique smooth bivector field
. Conversely, given any smooth bivector field
on
, the same formula
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)
*
satisfies
(hence it a Poisson bivector)
* the map
is a Lie algebra homomorphism, i.e. the Hamiltonian vector fields satisfy
* the graph
defines a Dirac structure, i.e. a Lagrangian subbundle
which is closed under the standard
Courant bracket In a field of mathematics known as differential geometry, the Courant bracket is a generalization of the Lie bracket from an operation on the tangent bundle to an operation on the direct sum of the tangent bundle and the vector bundle of ''p''-f ...
.
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
. 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 subspace
; when
, i.e. the map
is of constant rank, the Poisson structure
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
is a
regular distribution; it is easy to check that it is involutive, therefore, by
Frobenius theorem,
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
is
singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular homology
* SINGULAR, an open source Computer Algebra System (CAS)
* Singular or sounder, a group of boar, ...
, i.e. the vector subspaces
have different dimensions.
An integral submanifold for
is a path-connected submanifold
satisfying
for all
. Integral submanifolds of
are automatically regularly immersed manifolds, and maximal integral submanifolds of
are called the leaves of
.
Moreover, each leaf
carries a natural symplectic form
determined by the condition
for all
and
. Correspondingly, one speaks of the symplectic leaves of
. Moreover, both the space
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
splits locally around a point
as the product of a symplectic manifold
and a transverse Poisson submanifold
vanishing at
. More precisely, if
, there are local coordinates
such that the Poisson bivector
splits as the sum
where
. Note that, when the rank of
is maximal (e.g. the Poisson structure is nondegenerate), one recovers the classical
Darboux theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
for symplectic structures.
Examples
Trivial Poisson structures
Every manifold
carries the trivial Poisson structure
, equivalently described by the bivector
. Every point of
is therefore a zero-dimensional symplectic leaf.
Nondegenerate Poisson structures
A bivector field
is called nondegenerate if
is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as
symplectic manifolds .
Indeed, there is a bijective correspondence between nondegenerate bivector fields
and
nondegenerate 2-forms , given by
where
is encoded by
. Furthermore,
is Poisson precisely if and only if
is closed; in such case, the bracket becomes the canonical
Poisson bracket from Hamiltonian mechanics:
Non-degenerate Poisson structures have only one symplectic leaf, namely
itself, and their Poisson algebra
become a
Poisson ring.
Linear Poisson structures
A Poisson structure
on a vector space
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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. Indeed, the dual
of any finite-dimensional Lie algebra
carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (
Kostant-
Kirillov-
Souriau) structure:
where
and the derivatives
are interpreted as elements of the bidual
. Equivalently, the Poisson bivector can be locally expressed as
where
are coordinates on
and
are the associated
structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of
,
Conversely, any linear Poisson structure
on
must be of this form, i.e. there exists a natural Lie algebra structure induced on
whose Lie-Poisson bracket recovers
.
The symplectic leaves of the Lie-Poisson structure on
are the orbits of the
coadjoint action of
on
.
Fibrewise linear Poisson structures
The previous example can be generalised as follows. A Poisson structure on the total space of a vector bundle
is called fibrewise linear when the bracket of two smooth functions
, whose restrictions to the fibres are linear, is still linear when restricted to the fibres. Equivalently, the Poisson bivector field
is asked to satisfy
for any
, where
is the scalar multiplication
.
The class of vector bundles with linear Poisson structures coincides actually with that of (dual of)
Lie algebroids. Indeed, the dual
of any Lie algebroid
carries a fibrewise linear Poisson bracket, uniquely defined by
where
is the evaluation by
. Equivalently, the Poisson bivector can be locally expressed as
where
are coordinates around a point
,
are fibre coordinates on
, dual to a local frame
of
, and
and
are the structure function of
, i.e. the unique smooth functions satisfying
Conversely, any fibrewise linear Poisson structure
on
must be of this form, i.e. there exists a natural Lie algebroid structure induced on
whose Lie-Poisson backet recovers
.
The symplectic leaves of
are the cotangent bundles of the
algebroid orbits ; equivalently, if
is integrable to a Lie groupoid
, they are the connecting components of the orbits of the
cotangent groupoid .
For
one recovers linear Poisson structures, while for
the fibrewise linear Poisson structure is the nondegenerate one given by the canonical symplectic structure of
.
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,
is a 3-vector field, which is always zero in dimension 2.
*Given any Poisson bivector field
on a
3-dimensional manifold , the bivector field
, for any
, is automatically Poisson.
*The
Cartesian product of two Poisson manifolds
and
is again a Poisson manifold.
*Let
be a (regular)
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
of dimension
on
and
a closed foliation two-form for which the power
is nowhere-vanishing. This uniquely determines a regular Poisson structure on
by requiring the symplectic leaves of
to be the leaves
of
equipped with the induced symplectic form
.
*Let
be a
Lie group acting on a Poisson manifold
by Poisson diffeomorphisms. If the action is
free and
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, the
quotient manifold inherits a Poisson structure
from
(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
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
of the
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
where
is the Schouten-Nijenhuis bracket with
. Note that such a sequence can be defined for every bivector on
; the condition
is equivalent to
, i.e.
being Poisson.
Using the morphism
, one obtains a morphism from the
de Rham complex
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly ada ...
to the Poisson complex
, inducing a group homomorphism
. 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.
Poisson maps
A smooth map
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
for every
and smooth functions
* the bivector fields
and
are
-related, i.e.
* the Hamiltonian vector fields associated to every smooth function
are
-related, i.e.
* the differential
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
, with Poisson maps as morphisms. If a Poisson map
is also a diffeomorphism, then we call
a Poisson-diffeomorphism.
Examples
* Given the product Poisson manifold
, the canonical projections
, for
, are Poisson maps.
* The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map.
*Given two Lie algebras
and
, the dual of any Lie algebra homomorphism
induces a Poisson map
between their linear Poisson structures.
*Given two Lie algebroids
and
, the dual of any Lie algebroid morphism
over the identity induces a Poisson map
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
, whereas symplectic maps abound.
Symplectic realisations
A symplectic realisation on a Poisson manifold M consists of a symplectic manifold
together with a Poisson map
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
is a surjective submersion. Examples of (full) symplectic realisations include the following:
* For the trivial Poisson structure
, one takes the cotangent bundle
, with its
canonical symplectic structure, and the projection
.
* For a non-degenerate Poisson structure
one takes
itself and the identity
.
* For the Lie-Poisson structure on
, one takes the cotangent bundle
of a Lie group
integrating
and the dual map
of the differential at the identity of the (left or right) translation
.
A symplectic realisation
is called complete if, for any complete Hamiltonian vector field
, the vector field
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
induces a structure of
Lie algebroid In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought of ...
on its cotangent bundle
, also called the cotangent algebroid. The anchor map is given by
while the Lie bracket on
is defined as
Several notions defined for Poisson manifolds can be interpreted via its Lie 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
* a Poisson structure on
is regular precisely when the associated Lie algebroid
is
* the Poisson cohomology groups coincide with the Lie algebroid cohomology groups of
with coefficients in the trivial representation.
It is of crucial importance to notice that the Lie algebroid
is not always integrable to a Lie groupoid.
Symplectic groupoids
A is a
Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smo ...
together with a symplectic form
which is also multiplicative (i.e. compatible with the groupoid structure). Equivalently, the graph of
is asked to be a
Lagrangian submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
of
. Among the several consequences, the dimension of
is automatically twice the dimension of
.
A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure
such that the source map
and the target map
are, respectively, a Poisson map and an anti-Poisson map. Moreover, the Lie algebroid
is isomorphic to the cotangent algebroid
associated to the Poisson manifold
.
Conversely, if the cotangent bundle
of a Poisson manifold is integrable to some Lie groupoid
, then
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
for the symplectic groupoid integrating a given Poisson manifold
is called Poisson homotopy groupoid and is simply the
Weinstein groupoid of the cotangent algebroid
, consisting of the quotient of the
Banach space of a special class of
paths in
up to a suitable equivalent relation. Equivalently,
can be described as an infinite-dimensional
symplectic quotient In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the ac ...
.
Examples of integrations
* The trivial Poisson structure
is always integrable, the symplectic groupoid being the bundle of abelian (additive) groups
with the canonical symplectic form.
* A non-degenerate Poisson structure on
is always integrable, the symplectic groupoid being the pair groupoid
together with the symplectic form
(for
).
* A Lie-Poisson structure on
is always integrable, the symplectic groupoid being the (
coadjoint) action groupoid
, for
the
simply connected integration of
, together with the canonical symplectic form of
.
* A Lie-Poisson structure on
is integrable if and only if the Lie algebroid
is integrable to a Lie groupoid
, the symplectic groupoid being the cotangent groupoid
with the canonical symplectic form.
Submanifolds
A Poisson submanifold of
is an
immersed submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...
such that the immersion map
is a Poisson map. Equivalently, one asks that every Hamiltonian vector field
, for
, is tangent to
.
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
is a Poisson map transverse to a Poisson submanifold
of
, the submanifold
of
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
which is transverse to every symplectic leaf
and such that the intersection
is a symplectic submanifold of
. It follows that any Poisson transversal
inherits a canonical Poisson structure
from
. In the case of a nondegenerate Poisson manifold
(whose only symplectic leaf is
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 In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold.
The infinitesimal counterpart of a Poisson–Lie group is a L ...
*
Poisson supermanifold
*
Kontsevich quantization formula In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of ...
References
Books and surveys
*
*
*
* Previous version available o
*
*
* See also th
reviewby Ping Xu in the Bulletin of the AMS.
*
{{Manifolds
Differential geometry
Symplectic geometry
Smooth manifolds
Structures on manifolds