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

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

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: $\{fg,h\}=f\{g,h\}+g\{f,h\}$.

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 $\pi \in \Gamma \left(\bigwedge ^{{2}}TM\right)$ 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

${\displaystyle [\cdot ,\cdot ]:{{\mathfrak {X}}^{p}}(M)\times {{\mathfrak {X}}^{q}}(M)\to {{\mathfrak {X}}^{p+q-1}}(M)}$

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 $\pi ^{{\sharp }}:T^{{*}}M\to TM$. The rank of $\pi$ at a point $x\in M$ is then the rank of the induced linear mapping $\pi _{{x}}^{{\sharp }}$. 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 $T_{{x}}S={\pi ^{{\sharp }}}(T_{{x}}^{{\ast }}M)$ 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 $2r$ 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]

See also

• Nambu-Poisson manifold
• Poisson–Lie group
• Poisson supermanifold

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

• 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.
• 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.