Poisson manifold

From Wikipedia, the free encyclopedia
Jump to: navigation, search

A Poisson structure on a smooth manifold  M is a Lie bracket  \{ \cdot,\cdot \} (called a Poisson bracket in this special case) on the algebra  {C^{\infty}}(M) of smooth functions on  M , subject to the Leibniz Rule

 \{ f g,h \} = f \{ g,h \} + g \{ f,h \} .

Said in another manner, it is a Lie-algebra structure on the vector space of smooth functions on  M such that  X_{f} \stackrel{\text{df}}{=} \{ f,\cdot \}: {C^{\infty}}(M) \to {C^{\infty}}(M) is a vector field for each smooth function  f , which we call the Hamiltonian vector field associated to  f . 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  M be a smooth manifold. Let  {C^{\infty}}(M) denote the real algebra of smooth real-valued functions on  M , where multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on  M is an  \mathbb{R} -bilinear map

 \{ \cdot,\cdot \}: {C^{\infty}}(M) \times {C^{\infty}}(M) \to {C^{\infty}}(M)

satisfying the following three conditions:

The first two conditions ensure that  \{ \cdot,\cdot \} defines a Lie-algebra structure on  {C^{\infty}}(M) , while the third guarantees that for each  f \in {C^{\infty}}(M) , the adjoint  \{ f,\cdot \}: {C^{\infty}}(M) \to {C^{\infty}}(M) is a derivation of the commutative product on  {C^{\infty}}(M) , i.e., is a vector field  X_{f} . It follows that the bracket  \{ f,g \} of functions  f and  g is of the form  \{ f,g \} = \pi(df \wedge dg) , where  \pi \in \Gamma \left( \bigwedge^{2} T M \right) is a smooth bi-vector field.

Conversely, given any smooth bi-vector field  \pi on  M , the formula  \{ f,g \} = \pi(df \wedge dg) defines a bilinear skew-symmetric bracket  \{ \cdot,\cdot \} that automatically obeys Leibniz's rule. The condition that the ensuing  \{ \cdot,\cdot \} be a Poisson bracket — i.e., satisfy the Jacobi identity — can be characterized by the non-linear partial differential equation  [\pi,\pi] = 0 , where

 [\cdot,\cdot]: {\mathfrak{X}^{p}}(M) \times {\mathfrak{X}^{q}}(M) \to {\mathfrak{X}^{p + q}}(M)

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  \pi^{\sharp}: T^{*} M \to T M . The rank of  \pi at a point  x \in M is then the rank of the induced linear mapping  \pi^{\sharp}_{x} . Its image consists of the values  {X_{f}}(x) of all Hamiltonian vector fields evaluated at  x . A point  x \in M is called regular for a Poisson structure  \pi on  M if and only if the rank of  \pi is constant on an open neighborhood of  x \in M ; otherwise, it is called a singular point. Regular points form an open dense subspace  M_{\mathrm{reg}} \subseteq M ; when  M_{\mathrm{reg}} = M , we call the Poisson structure itself regular.

An integral sub-manifold for the (singular) distribution  {\pi^{\sharp}}(T^{*} M) is a path-connected sub-manifold  S \subseteq M satisfying  T_{x} S = {\pi^{\sharp}}(T^{\ast}_{x} M) for all  x \in S . Integral sub-manifolds of  \pi are automatically regularly immersed manifolds, and maximal integral sub-manifolds of  \pi are called the leaves of  \pi . Each leaf  S carries a natural symplectic form  \omega_{S} \in {\Omega^{2}}(S) determined by the condition  [{\omega_{S}}(X_{f},X_{g})](x) = - \{ f,g \}(x) for all  f,g \in {C^{\infty}}(M) and  x \in S . Correspondingly, one speaks of the symplectic leaves of  \pi .[3] Moreover, both the space  M_{\mathrm{reg}} of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

Examples[edit]

  • Every manifold  M carries the trivial Poisson structure  \{ f,g \} = 0 .
  • Every symplectic manifold  (M,\omega) is Poisson, with the Poisson bi-vector  \pi equal to the inverse  \omega^{-1} of the symplectic form  \omega .
  • The dual  \mathfrak{g}^{*} of a Lie algebra  (\mathfrak{g},[\cdot,\cdot]) is a Poisson manifold. A coordinate-free description can be given as follows:  \mathfrak{g} naturally sits inside  {C^{\infty}}(\mathfrak{g}^{*}) , and the rule  \{ X,Y \} \stackrel{\text{df}}{=} [X,Y] for each  X,Y \in \mathfrak{g} induces a linear Poisson structure on  \mathfrak{g}^{*} , i.e., one for which the bracket of linear functions is again linear. Conversely, any linear Poisson structure must be of this form.
  • Let  \mathcal{F} be a (regular) foliation of dimension  2 r on  M and  \omega \in {\Omega^{2}}(\mathcal{F}) a closed foliation two-form for which  \omega^{r} is nowhere-vanishing. This uniquely determines a regular Poisson structure on  M by requiring that the symplectic leaves of  \pi be the leaves  S of  \mathcal{F} equipped with the induced symplectic form  \omega|_S .

Poisson Maps[edit]

If  (M,\{ \cdot,\cdot \}_{M}) and  (M',\{ \cdot,\cdot \}_{M'}) are two Poisson manifolds, then a smooth mapping  \varphi: M \to M' is called a Poisson map if it respects the Poisson structures, namely, if for all  x \in M and smooth functions  f,g \in {C^{\infty}}(M') , we have:

 {\{ f,g \}_{M'}}(\varphi(x)) = {\{ f \circ \varphi,g \circ \varphi \}_{M}}(x).

In terms of Poisson bi-vectors, the condition that a map be Poisson is tantamount to requiring that  \pi_{M} and  \pi_{M'} be  \varphi -related.

Poisson manifolds are the objects of a category  \mathfrak{Poiss} , with Poisson maps as morphisms.

Examples of Poisson maps:

  • The Cartesian product  (M_{0} \times M_{1},\pi_{0} \times \pi_{1}) of two Poisson manifolds  (M_{0},\pi_{0}) and  (M_{1},\pi_{1}) is again a Poisson manifold, and the canonical projections  \mathrm{pr}_{i}: M_{0} \times M_{1} \to M_{i} , for  i \in \{ 0,1 \} , 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  \mathbb{R}^{2} \to \mathbb{R}^{4} , 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]

References[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. 
  • 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.