Poisson manifold

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

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:

• Skew symmetry: $\{ f,g \} = - \{ g,f \}$.
• Jacobi identity: $\{ f,\{ g,h \} \} + \{ g,\{ h,f \} \} + \{ h,\{ f,g \} \} = 0$.
• Leibniz's Rule: $\{ f g,h \} = f \{ g,h \} + g \{ f,h \}$.

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

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

• 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

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]

References

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.