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.
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.
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 on , i.e. satisfying the following three conditions:
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 some local coordinates , any Poisson bracket is given by
A Poisson bivector on a smooth manifold is a bivector field satisfying the non-linear partial differential equation , where
denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing some local coordinates , any Poisson bivector is given by
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.
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, 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
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
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.
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
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 .
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.
- The Cartesian product of two Poisson manifolds and is again a Poisson manifold.
- Let be a (regular) foliation 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 that the symplectic leaves of 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, the quotient manifold inherits a Poisson structure from (namely, it is the only one such that the submersion is a Poisson map).
The Poisson cohomology groups of a Poisson manifold are the cohomology groups of the cochain complex
Using the morphism , one obtains a morphism from the de Rham complex 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.
A smooth mapping between Poisson manifolds is called a Poisson map 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.
- 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 the linear Poisson structures.
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.
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 on its cotangent bundle . The anchor map is given by while the Lie bracket on is defined as
- 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.
A symplectic groupoid is a Lie groupoid 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 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 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.
A Poisson submanifold of is an immersed submanifold 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.
- Lichnerowicz, A. (1977). "Les variétés de Poisson et leurs algèbres de Lie associées". J. Diff. Geom. 12 (2): 253–300. doi:10.4310/jdg/1214433987. MR 0501133.
- Weinstein, Alan (1983-01-01). "The local structure of Poisson manifolds". Journal of Differential Geometry. 18 (3). doi:10.4310/jdg/1214437787. ISSN 0022-040X.
- Karasev, M V (1987-06-30). "Analogues of the Objects of Lie Group Theory for Nonlinear Poisson Brackets". Mathematics of the USSR-Izvestiya. 28 (3): 497–527. doi:10.1070/im1987v028n03abeh000895. ISSN 0025-5726.
- Crainic, Marius; Marcut, Ioan (2011). "On the extistence of symplectic realizations". Journal of Symplectic Geometry. 9 (4): 435–444. doi:10.4310/JSG.2011.v9.n4.a2. ISSN 1540-2347.
- Crainic, Marius; Fernandes, Rui Loja (2004-01-01). "Integrability of Poisson Brackets". Journal of Differential Geometry. 66 (1). doi:10.4310/jdg/1090415030. ISSN 0022-040X.
- Weinstein, Alan (1987-01-01). "Symplectic groupoids and Poisson manifolds". Bulletin of the American Mathematical Society. 16 (1): 101–105. doi:10.1090/S0273-0979-1987-15473-5. ISSN 0273-0979.
- Albert, Claude; Dazord, Pierre (1991). Dazord, Pierre; Weinstein, Alan (eds.). "Groupoïdes de Lie et Groupoïdes Symplectiques". Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications (in French). New York, NY: Springer US. 20: 1–11. doi:10.1007/978-1-4613-9719-9_1. ISBN 978-1-4613-9719-9.
- Liu, Z. -J.; Xu, P. (1996-01-01). "Exact lie bialgebroids and poisson groupoids". Geometric & Functional Analysis GAFA. 6 (1): 138–145. doi:10.1007/BF02246770. ISSN 1420-8970. S2CID 121836719.
- Crainic, Marius; Fernandes, Rui (2003-03-01). "Integrability of Lie brackets". Annals of Mathematics. 157 (2): 575–620. doi:10.4007/annals.2003.157.575. ISSN 0003-486X.
- Zambon, Marco (2011). Ebeling, Wolfgang; Hulek, Klaus; Smoczyk, Knut (eds.). "Submanifolds in Poisson geometry: a survey". Complex and Differential Geometry. Springer Proceedings in Mathematics. Berlin, Heidelberg: Springer. 8: 403–420. doi:10.1007/978-3-642-20300-8_20. ISBN 978-3-642-20300-8.
Books and surveys
- Bhaskara, K. H.; Viswanath, K. (1988). Poisson algebras and Poisson manifolds. Longman. ISBN 0-582-01989-3.
- Cannas da Silva, Ana; Weinstein, Alan (1999). Geometric models for noncommutative algebras. AMS Berkeley Mathematics Lecture Notes, 10.
- Dufour, J.-P.; Zung, N.T. (2005). Poisson Structures and Their Normal Forms. 242. Birkhäuser Progress in Mathematics.
- Crainic, Marius; Loja Fernandes, Rui; Mărcuț, Ioan (2021). Lectures on Poisson Geometry (PDF). Graduate Studies in Mathematics. American Mathematical Society (to appear). Previous version available on .
- Guillemin, Victor; Sternberg, Shlomo (1984). Symplectic Techniques in Physics. New York: Cambridge University Press. ISBN 0-521-24866-3.
- Libermann, Paulette; Marle, C.-M. (1987). Symplectic geometry and analytical mechanics. Dordrecht: Reidel. ISBN 90-277-2438-5.
- Vaisman, Izu (1994). Lectures on the Geometry of Poisson Manifolds. Birkhäuser. See also the review by Ping Xu in the Bulletin of the AMS.
- Weinstein, Alan (1998). "Poisson geometry". Differential Geometry and Its Applications. 9 (1–2): 213–238. doi:10.1016/S0926-2245(98)00022-9.