Poisson manifold

In geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule

.

Said in another manner, it is a Lie algebra structure on the vector space of smooth functions on such that is a vector field for each smooth function , which we call the Hamiltonian vector field associated to . These vector fields span a completely integrable singular foliation, each of whose maximal integral sub-manifolds inherits a symplectic structure. One may thus informally view a Poisson structure on a smooth manifold as a smooth partition of the ambient manifold into even-dimensional symplectic leaves, which are not necessarily of the same dimension.

Poisson structures are one instance of Jacobi structures introduced by André Lichnerowicz in 1977.[1] They were further studied in the classical paper of Alan Weinstein,[2] 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.

DefinitionEdit

Let   be a smooth manifold. Let   denote the real algebra of smooth real-valued functions on  , where multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on   is an  -bilinear map

 

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 adjoint   is a derivation of the commutative product on  , i.e., is a vector field  . It follows that the bracket   of functions   and   is of the form

 ,

where   is a smooth bi-vector field, called the Poisson bi-vector.

Conversely, given any smooth bi-vector field   on  , the formula   defines a bilinear skew-symmetric bracket   that automatically obeys Leibniz's rule. The condition that the ensuing   be a Poisson bracket — i.e., satisfy the Jacobi identity — can be characterized by the non-linear partial differential equation  , where

 

denotes the Schouten–Nijenhuis bracket on multi-vector fields. It is customary and convenient to switch between the bracket and the bi-vector points of view, and we shall do so below.

Symplectic leavesEdit

A Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds, called its symplectic leaves.

Note that a bi-vector field can be regarded as a skew homomorphism  . The rank of   at a point   is then the rank of the induced linear mapping  . Its image consists of the values   of all Hamiltonian vector fields evaluated at  . 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  , we call the Poisson structure itself regular.

An integral sub-manifold for the (singular) distribution   is a path-connected sub-manifold   satisfying   for all  . Integral sub-manifolds of   are automatically regularly immersed manifolds, and maximal integral sub-manifolds of   are called the leaves of  . Each leaf   carries a natural symplectic form   determined by the condition   for all   and  . Correspondingly, one speaks of the symplectic leaves of  .[3] Moreover, both the space   of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

ExamplesEdit

  • Every manifold   carries the trivial Poisson structure  .
  • Every symplectic manifold   is Poisson, with the Poisson bi-vector   equal to the inverse   of the symplectic form  .
  • The dual   of a Lie algebra   is a Poisson manifold. A coordinate-free description can be given as follows:   naturally sits inside  , and the rule   for each   induces a linear Poisson structure on  , i.e., one for which the bracket of linear functions is again linear. Conversely, any linear Poisson structure must be of this form.
  • Let   be a (regular) foliation of dimension   on   and   a closed foliation two-form for which   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  .

Poisson mapsEdit

If   and   are two Poisson manifolds, then a smooth mapping   is called a Poisson map if it respects the Poisson structures, namely, if for all   and smooth functions  , we have:

 

If   is also a diffeomorphism, then we call   a Poisson-diffeomorphism. In terms of Poisson bi-vectors, the condition that a map be Poisson is equivalent to requiring that   and   be  -related.

Poisson manifolds are the objects of a category  , with Poisson maps as morphisms.

Examples of Poisson maps:

  • The Cartesian product   of two Poisson manifolds   and   is again a Poisson manifold, and the canonical projections  , for  , are Poisson maps.
  • The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map.

It must be highlighted that the notion of a Poisson map is fundamentally different from that of a symplectic map. For instance, with their standard symplectic structures, there do not exist Poisson maps  , whereas symplectic maps abound.

One interesting, and somewhat surprising, fact is that any Poisson manifold is the codomain/image of a surjective, submersive Poisson map from a symplectic manifold. [4][5][6]

See alsoEdit

NotesEdit

  1. ^ 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.
  2. ^ Weinstein, Alan (1983). "The local structure of Poisson manifolds". Journal of Differential Geometry. 18 (3): 523–557.
  3. ^ Fernandes, R.L.; Marcut, I. (2014). Lectures on Poisson Geometry. Yet unpublished lecture notes.[1]
  4. ^ Crainic, Marius; Marcut, I. (2011). "On the existence of symplectic realizations". J. Symplectic Geom. 9 (4): 435–444.
  5. ^ Karasev, M. (1987). "Analogues of objects of Lie group theory for nonlinear Poisson brackets". Math. USSR Izv. 28: 497–527.
  6. ^ Weinstein, A. (1983). "The local structure of Poisson manifolds". J. Diff. Geom. 18 (3): 523–557.

ReferencesEdit