Poisson manifold

In differential geometry, a Poisson structure on a smooth manifold formula_1 is a Lie bracket formula_2 (called a Poisson bracket in this special case) on the algebra formula_3 of smooth functions on formula_1, subject to the Leibniz rule
Equivalently, formula_2 defines a Lie algebra structure on the vector space formula_3 of smooth functions on formula_1 such that formula_9 is a vector field for each smooth function formula_10 (making formula_3 into a Poisson algebra).
Poisson structures have been 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.
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 formula_1 be a smooth manifold and let formula_3 denote the real algebra of smooth real-valued functions on formula_1, where the multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on formula_1 is an formula_16-bilinear map
defining a structure of Poisson algebra on formula_3, i.e. satisfying the following three conditions:
The first two conditions ensure that formula_2 defines a Lie-algebra structure on formula_3, while the third guarantees that, for each formula_24, the linear map formula_25 is a derivation of the algebra formula_3, i.e., it defines a vector field formula_27 called the Hamiltonian vector field associated to formula_10. 
Choosing some local coordinates formula_29, any Poisson bracket is given byformula_30for formula_31 the Poisson bracket of the coordinate functions. 
As bivector.
A Poisson bivector on a smooth manifold formula_1 is a bivector field formula_33 satisfying the non-linear partial differential equation formula_34, where 
denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing some local coordinates formula_29, any Poisson bivector is given byformula_37for formula_38 skew-symmetric smooth functions on formula_39.
Equivalence of the definitions.
Let formula_2 be a bilinear skew-symmetric bracket satisfying Leibniz's rule; then the function formula_41 can be described as 
for a unique smooth bivector field formula_43. Conversely, given any smooth bivector field formula_44 on formula_1, the same formula formula_42 defines a bilinear skew-symmetric bracket formula_2 that automatically obeys Leibniz's rule.
Last, the following conditions are equivalent
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 formula_55. The image formula_56 consists therefore of the values formula_57 of all Hamiltonian vector fields evaluated at every formula_58. 
The rank of formula_44 at a point formula_58 is the rank of the induced linear mapping formula_61. A point formula_58 is called regular for a Poisson structure formula_44 on formula_1 if and only if the rank of formula_44 is constant on an open neighborhood of formula_58; otherwise, it is called a singular point. Regular points form an open dense subspace formula_67; when formula_68, i.e. the map formula_69 is of constant rank, the Poisson structure formula_44 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 formula_56 is a regular distribution; it is easy to check that it is involutive, therefore, by Frobenius theorem, formula_1 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 formula_56 is singular, i.e. the vector subspaces formula_74 have different dimensions.
An integral submanifold for formula_75 is a path-connected submanifold formula_76 satisfying formula_77 for all formula_78. Integral submanifolds of formula_44 are automatically regularly immersed manifolds, and maximal integral submanifolds of formula_44 are called the leaves of formula_44.
Moreover, each leaf formula_82 carries a natural symplectic form formula_83 determined by the condition formula_84 for all formula_85 and formula_78. Correspondingly, one speaks of the symplectic leaves of formula_44. Moreover, both the space formula_88 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 formula_89 splits locally around a point formula_90 as the product of a symplectic manifold formula_91 and a transverse Poisson submanifold formula_92 vanishing at formula_93. More precisely, if formula_94, there are local coordinates formula_95 such that the Poisson bivector formula_44 splits as the sumformula_97where formula_98. Note that, when the rank of formula_44 is maximal (e.g. the Poisson structure is nondegenerate), one recovers the classical Darboux theorem for symplectic structures.
Examples.
Trivial Poisson structures.
Every manifold formula_1 carries the trivial Poisson structure formula_101, equivalently described by the bivector formula_102. Every point of formula_1 is therefore a zero-dimensional symplectic leaf.
Nondegenerate Poisson structures.
A bivector field formula_44 is called nondegenerate if formula_105 is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as symplectic manifolds formula_106.
Indeed, there is a bijective correspondence between nondegenerate bivector fields formula_44 and nondegenerate 2-forms formula_108, given byformula_109where formula_108 is encoded by formula_111. Furthermore, formula_44 is Poisson precisely if and only if formula_108 is closed; in such case, the bracket becomes the canonical Poisson bracket from Hamiltonian mechanics:formula_114Non-degenerate Poisson structures have only one symplectic leaf, namely formula_1 itself, and their Poisson algebra formula_116 become a Poisson ring.
Linear Poisson structures.
A Poisson structure formula_117 on a vector space formula_118 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 formula_119 of any finite-dimensional Lie algebra formula_120 carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (Kostant-Kirillov-Souriau) structure:
formula_121,
where formula_122 and the derivatives formula_123 are interpreted as elements of the bidual formula_124. Equivalently, the Poisson bivector can be locally expressed asformula_125where formula_126 are coordinates on formula_119 and formula_128 are the associated structure constants of formula_129, 
Conversely, any linear Poisson structure formula_117 on formula_118 must be of this form, i.e. there exists a natural Lie algebra structure induced on formula_132 whose Lie-Poisson bracket recovers formula_117.
The symplectic leaves of the Lie-Poisson structure on formula_134 are the orbits of the coadjoint action of formula_135 on formula_134.
Poisson cohomology.
The Poisson cohomology groups formula_157 of a Poisson manifold are the cohomology groups of the cochain complexformula_158
where formula_159 is the Schouten-Nijenhuis bracket with formula_44. Note that such a sequence can be defined for every bivector on formula_1; the condition formula_162 is equivalent to formula_163, i.e. formula_1 being Poisson.
Using the morphism formula_105, one obtains a morphism from the de Rham complex formula_166 to the Poisson complex formula_167, inducing a group homomorphism formula_168. 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: 
Poisson maps.
A smooth mapping formula_173 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):
An anti-Poisson map satisfies analogous conditions with a minus sign on one side. 
Poisson manifolds are the objects of a category formula_187, with Poisson maps as morphisms. If a Poisson map formula_188 is also a diffeomorphism, then we call formula_189 a Poisson-diffeomorphism. 
Examples.
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 formula_197, whereas symplectic maps abound.
Symplectic realisations.
A symplectic realisation on a Poisson manifold M consists of a symplectic manifold formula_198 together with a Poisson map formula_199 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 formula_200 is a surjective submersion. Examples of (full) symplectic realisations include the following:
A symplectic realisation formula_200 is called complete if, for any complete Hamiltonian vector field formula_214, the vector field formula_215 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 formula_152 induces a structure of Lie algebroid on its cotangent bundle formula_203. The anchor map is given by formula_105 while the Lie bracket on formula_219 is defined asformula_220Several notions defined for Poisson manifolds can be interpreted via its Lie algebroid formula_202:
It is of crucial importance to notice that the Lie algebroid formula_202 is not always integrable to a Lie groupoid.
Symplectic groupoids.
A is a Lie groupoid formula_226 together with a symplectic form formula_227 which is also multiplicative (i.e. compatible with the groupoid structure). Equivalently, the graph of formula_108 is asked to be a Lagrangian submanifold of formula_229.
A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure formula_44 such that the source map formula_231 and the target map formula_232 are, respectively, a Poisson map and an anti-Poisson map. Moreover, the Lie algebroid formula_233 is isomorphic to the cotangent algebroid formula_202 associated to the Poisson manifold formula_152. Conversely, if the cotangent bundle formula_202 of a Poisson manifold is integrable to some Lie groupoid formula_226, then formula_238 is automatically a symplectic groupoid.
Accordingly, the integrability problem for a Poisson manifold consists in finding a symplectic 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.
Submanifolds.
A Poisson submanifold of formula_239 is an immersed submanifold formula_240 such that the immersion map formula_241 is a Poisson map. Equivalently, one asks that every Hamiltonian vector field formula_242, for formula_243, is tangent to formula_244.
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:
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 formula_250 which is transverse to every symplectic leaf formula_82 and such that the intersection formula_252 is a symplectic submanifold of formula_253. It follows that any Poisson transversal formula_254 inherits a canonical Poisson structure formula_255 from formula_44. In the case of a nondegenerate Poisson manifold formula_239 (whose only symplectic leaf is formula_1 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.