Poisson manifold

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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[edit]

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.

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[edit]

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[edit]

  • 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 Maps[edit]

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:

  • 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 also[edit]

Notes[edit]

  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. [1]
  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[edit]

  • Bhaskara, K. H.; Viswanath, K. (1988). Poisson algebras and Poisson manifolds. Longman. ISBN 0-582-01989-3. 
  • Cannas da Silva, A.; Weinstein, A. (1999). Geometric models for noncommutative algebras. AMS Berkeley Mathematics Lecture Notes, 10. 
  • Crainic, M.; Fernandes, R.L. (2004). "Integrability of Poisson Brackets". J. Diff. Geom. 66 (1): 71–137. 
  • Crainic, M.; Marcut, I. (2011). "On the existence of symplectic realizations". J. Symplectic Geom. 9 (4): 435–444. 
  • Dufour, J.-P.; Zung, N.T. (2005). Poisson Structures and Their Normal Forms 242. Birkhäuser Progress in Mathematics. 
  • Fernandes, R.L.; Marcut, I. (2014). Lectures on Poisson Geometry. Yet unpublished lecture notes. [2]
  • Guillemin, V.; Sternberg, S. (1984). Symplectic Techniques in Physics. New York: Cambridge Univ. Press. ISBN 0-521-24866-3. 
  • Karasev, M. (1987). "Analogues of objects of Lie group theory for nonlinear Poisson brackets". Math.USSR Izv. 28: 497–527. 
  • Kirillov, A. A. (1976). "Local Lie algebras". Russ. Math. Surv. 31 (4): 55–75. doi:10.1070/RM1976v031n04ABEH001556. 
  • Libermann, P.; Marle, C.-M. (1987). Symplectic geometry and analytical mechanics. Dordrecht: Reidel. ISBN 90-277-2438-5. 
  • Lichnerowicz, A. (1977). "Les variétés de Poisson et leurs algèbres de Lie associées". J. Diff. Geom. 12 (2): 253–300. MR 0501133. 
  • Marcut, I. (2013). Normal forms in Poisson geometry. PhD Thesis: Utrecht University.  Available at thesis
  • Vaisman, I. (1994). Lectures on the Geometry of Poisson Manifolds. Birkhäuser.  See also the review by Ping Xu in the Bulletin of the AMS.
  • Weinstein, A. (1983). "The local structure of Poisson manifolds". J. Diff. Geom. 18 (3): 523–557. 
  • Weinstein, A. (1998). "Poisson geometry". Differential Geometry and its Applications 9 (1-2): 213–238.