Poisson manifold

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A Poisson structure on a smooth manifold  M is a Lie bracket  \{ \cdot,\cdot \} (called a Poisson bracket) on the algebra  {C^{\infty}}(M) of smooth functions on  M , subject to the Leibniz-type condition  \{fg,h\}=f\{g,h\}+g\{f,h\} . Otherwise said, it is a Lie algebra structure on smooth functions, for which each  \{ f,\cdot\} acts as a vector field  X_f , called the Hamiltonian vector field associated to the function  f . Such vector fields span a completely integrable singular foliation, each of whose maximal integral submanifolds inherits a symplectic structure. One may thus informally view a Poisson structure on a manifold as a smooth partition of the ambient manifold into symplectic leaves, not necessarily of the same dimension.

Poisson structures are one instance of the 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 in 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, and  \{ \cdot,\cdot\}: {C^{\infty}}(M) stand for the algebra (over real numbers) of smooth, real-valued functions on  M , under pointwise multiplication. 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) —that is, is a vector field  X_f . It follows that the bracket  \{f,g\} of functions  f,g is of the form  \{f,g\}=\pi(df \wedge dg) , where  \pi \in \Gamma\left(\bigwedge^2TM \right) is a bivector field.

Conversely, given any smooth bivector field  \pi on  M , formula  \{f,g\}=\pi(df \wedge dg) define a bilinear, skew-symmetric bracket  \{ \cdot,\cdot \} which automatically obeys Leibniz's rule. The condition that the ensuing  \{ \cdot,\cdot \} be a Poisson bracket—that is, that it satisfy the Jacobi identity—can be characterized by the non-linear, partial differential equation  [\pi,\pi]=0 , where here  [\cdot,\cdot]:\mathfrak{X}^p(M) \times \mathfrak{X}^q(M) \to \mathfrak{X}^{p+q-1}(M) denotes the Schouten–Nijenhuis bracket on multivector fields. It is customary and convenient to switch between the bracket and the bivector points of view, and we shall do so below.

Symplectic Leaves[edit]

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

Note that a bivector field can be regarded as a skew homomorphism  \pi^{\sharp}: T^{\ast}M \to TM. The rank of  \pi at a point  x \in M is the rank of the induced linear map  \pi^{\sharp}_x. Its image consists of all values of Hamiltonian vector fields X_f(x) at  x . A point  x \in M is called regular for a Poisson structure  \pi on  M if the rank of  \pi is constant in an open around  x \in M ; otherwise it is called a singular point. Regular points form an open, dense subspace  M_{\mathrm{reg}} \subset M; when  M_{\mathrm{reg}} = M we call the Poisson structure itself regular.

An integral submanifold for the (singular) distribution \pi^{\sharp}(T^{\ast}M) is a path-connected submanifold  S \subset M satisfying  T_xS=\pi^{\sharp}T^{\ast}_xM for all  x \in S. Integral submanifold of  \pi are automatically regularly immersed manifolds, and maximal integral submanifolds 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 symplectic leaves of  \pi .[3] Moreover, both the space  M_{\mathrm{reg}} of regular points and its complement are saturated by symplectic leaves; thus 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 Poisson bivector the inverse  \pi:=\omega^{-1} of the symplectic form  \omega .
  • The dual \mathfrak{g}^{\ast} 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}^{\ast}), and the rule \{X,Y\}:=[X,Y] for each X,Y \in \mathfrak{g} induces a linear Poisson structure on \mathfrak{g}^{\ast}—that is, 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 2r on M, and  \omega \in \Omega^2(\mathcal{F}) a closed, foliated two-form 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 Poisson manifolds, a smooth map  \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 bivectors, 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 in 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 are Poisson maps.
  • The inclusion of a symplectic leaf, or of an open subspace, are Poisson maps.

It must be highlighted that the notion of 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 insteresting, and somewhat surprising fact is that any Poisson manifold is the codomain/image of a surjective, submersive Poisson map from a symplectic manifold, cf.[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.