Poisson manifold

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.

Definition

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:

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.

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 Leaves

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.

Examples

Poisson Maps

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:

In terms of Poisson bi-vectors, the condition that a map be Poisson is tantamount to requiring that and be -related.

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

Examples of Poisson maps:

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 also

Notes

  1. Lichnerowicz, A. (1977). "Les variétés de Poisson et leurs algèbres de Lie associées". J. Diff. Geom. 12 (2): 253–300. MR 0501133.
  2. Weinstein, A. (1983). "The local structure of Poisson manifolds". J. Diff. Geom. 18 (3): 523–557.
  3. Fernandes, R.L.; Marcut, I. (2014). Lectures on Poisson Geometry. Yet unpublished lecture notes.
  4. Crainic, M.; 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.

References

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