Poisson manifold

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In mathematics, a Poisson manifold is a smooth manifold $M$ equipped with a bilinear map $\{\cdot ,\cdot \}_{{M}}$ (called a Poisson bracket) on the algebra ${C^{{\infty }}}(M)$ of smooth functions on $M$ such that $({C^{{\infty }}}(M),\{\cdot ,\cdot \}_{{M}})$ is a Poisson algebra. One usually denotes a Poisson manifold by the ordered pair $(M,\{\cdot ,\cdot \}_{{M}})$ . Since their introduction by André Lichnerowicz in 1977,^[1] the subjects of Poisson geometry and the cohomology of Poisson manifolds have developed into a wide field of research, which includes modern-day non-commutative geometry.

It is a fact that every symplectic manifold is a Poisson manifold but not vice-versa. This will be explained in Section 2.

Definition

A Poisson bracket (or Poisson structure) on a smooth manifold $M$ is a bilinear map

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

that satisfies the following three properties:

It is skew-symmetric: $\{f,g\}=-\{g,f\}$ .
It obeys the Jacobi Identity: $\{f,\{g,h\}\}+\{g,\{h,f\}\}+\{h,\{f,g\}\}=0$ .
It obeys Leibniz's Rule with respect to the first argument: $\{fg,h\}=f\{g,h\}+g\{f,h\}$ .

By skew-symmetry, the Poisson bracket automatically satisfies Leibniz's Rule with respect to the second argument. The last property basically states that the map $f\mapsto \{f,g\}$ is a derivation on ${C^{{\infty }}}(M)$ for any fixed $g\in {C^{{\infty }}}(M)$ . Every derivation $\delta$ on ${C^{{\infty }}}(M)$ can be written as a directional derivative $[\delta (f)](x)={({\mathrm {d}}f)_{{x}}}((X_{{\delta }})_{{x}})$ , where $x\in M$ , for some vector field $X_{{\delta }}$ . It follows that for $g\in {C^{{\infty }}}(M)$ , we obtain a vector field $X_{{g}}$ such that $\{f,g\}(x)={({\mathrm {d}}f)_{{x}}}((X_{{g}})_{{x}})$ , where $x\in M$ (written more briefly, $\{f,g\}={{\mathrm {d}}f}(X_{{g}})$ ). The vector field $X_{{g}}$ is called the Hamiltonian vector field corresponding to $g$ . Notice that

$\langle {\mathrm {d}}f,X_{{g}}\rangle ={X_{{g}}}(f)=\{f,g\}=-{X_{{f}}}(g)=-\langle {\mathrm {d}}g,X_{{f}}\rangle$ ,

where $\langle \cdot ,\cdot \rangle$ is the pairing between the cotangent and tangent bundles of $M$ . Therefore, $\{f,g\}$ depends only on the differentials ${\mathrm {d}}f$ and ${\mathrm {d}}g$ . Any Poisson bracket yields a map from the cotangent bundle to the tangent bundle that sends ${\mathrm {d}}f$ to $X_{{f}}$ .

The Poisson Bivector

Given a Poisson manifold $(M,\{\cdot ,\cdot \}_{{M}})$ , the pairing between the cotangent and tangent bundles yields a bivector field $\eta$ on $M$ , called the Poisson bivector field. The Poisson bivector field is a contravariant skew-symmetric 2-tensor field $\eta \in \Gamma \left(\bigwedge ^{{2}}TM\right)$ that satisfies the following:

$\forall x\in M:\quad {\{f,g\}_{{M}}}(x)=\langle ({\mathrm {d}}f)_{{x}}\otimes ({\mathrm {d}}g)_{{x}},\eta _{{x}}\rangle$ .

Conversely, given a smooth bivector field $\eta$ on $M$ , we can use the formula above to define a skew-symmetric bracket $\{\cdot ,\cdot \}_{{\eta }}$ that obeys Leibniz's rule with respect to each argument. However, we cannot claim that $\{\cdot ,\cdot \}_{{\eta }}$ is a Poisson bracket because the Jacobi Identity may not hold (in this case, we call $\{\cdot ,\cdot \}_{{\eta }}$ an almost-Poisson structure). Indeed, $\{\cdot ,\cdot \}_{{\eta }}$ is a Poisson bracket if and only if the Schouten–Nijenhuis bracket $[\eta ,\eta ]$ equals zero.

In terms of local coordinates, the bivector field at a point $x=(x_{{1}},\ldots ,x_{{m}})$ can be expressed as

$\forall x\in M:\quad \eta _{x}=\sum _{{i,j=1}}^{{m}}{\eta _{{i,j}}}(x)\cdot \left({\frac {\partial }{\partial x_{{i}}}}\right)_{{x}}\otimes \left({\frac {\partial }{\partial x_{{j}}}}\right)_{{x}}$ ,

so that

$\{f,g\}(x)=\langle ({\mathrm {d}}f)_{{x}}\otimes ({\mathrm {d}}g)_{{x}},\eta _{{x}}\rangle =\sum _{{i,j=1}}^{{m}}{\eta _{{i,j}}}(x)\left({\frac {\partial f}{\partial x_{{i}}}}\right)_{{x}}\left({\frac {\partial g}{\partial x_{{j}}}}\right)_{{x}}$ .

For a symplectic manifold $(M,\omega )$ , we can define a bivector field $\eta$ on $M$ using the pairing between the cotangent and tangent bundles given by the symplectic form $\omega$ . This pairing is well-defined because $\omega$ is nondegenerate. Hence, the difference between a symplectic manifold and a Poisson manifold is that the symplectic form is regular (of full rank) everywhere but the Poisson bivector field need not have full rank everywhere. When the Poisson bivector field is zero everywhere, we call it the trivial Poisson structure.

Poisson Maps

A Poisson map from a Poisson manifold $(M,\{\cdot ,\cdot \}_{{M}})$ to another Poisson manifold $(N,\{\cdot ,\cdot \}_{{N}})$ is defined to be a smooth map $\varphi :M\to N$ that respects the Poisson structures in the following sense:

$\{f_{{1}},f_{{2}}\}_{{N}}\circ \varphi =\{f_{{1}}\circ \varphi ,f_{{2}}\circ \varphi \}_{{M}}$ .

A Poisson map may be viewed as a morphism in the category of Poisson manifolds.

The Product of Poisson Manifolds

Given two Poisson manifolds $(M,\{\cdot ,\cdot \}_{{M}})$ and $(N,\{\cdot ,\cdot \}_{{N}})$ , a Poisson bracket may be defined on the product manifold $M\times N$ . Letting $f_{{1}}$ and $f_{{2}}$ be two smooth functions defined on $M\times N$ , one can define a new Poisson bracket $\{\cdot ,\cdot \}_{{M\times N}}$ in terms of $\{\cdot ,\cdot \}_{{M}}$ and $\{\cdot ,\cdot \}_{{N}}$ as follows:

${\{f_{{1}},f_{{2}}\}_{{M\times N}}}(x,y)={\{{f_{{1}}}(x,\cdot ),{f_{{2}}}(x,\cdot )\}_{{N}}}(y)+{\{{f_{{1}}}(\cdot ,y),{f_{{2}}}(\cdot ,y)\}_{{M}}}(x)$ ,

where $x\in M$ and $y\in N$ are to be held constant. In other words, if

$f(\cdot ,\cdot ):M\times N\to {\mathbf {R}}$ ,

then both

$f(x,\cdot ):N\to {\mathbf {R}}$

and

$f(\cdot ,y):M\to {\mathbf {R}}$

are implied.

The Symplectic Leaves of a Poisson Structure

A Poisson manifold $(M,\{\cdot ,\cdot \}_{{M}})$ can be split into a collection of symplectic leaves. This splitting arises from the foliation of disjoint regions of $M$ where the Poisson bivector field has constant rank. Each leaf of the foliation is thus an even-dimensional sub-manifold of $M$ that is itself a symplectic manifold. Distinct symplectic leaves may have different dimensions. Two points lie in the same leaf if and only if they are joined by a piecewise-smooth curve where each piece is the integral curve of a Hamiltonian vector field. The relation "piecewise-connected by integral curves of Hamiltonian fields" is an equivalence relation on $M$ , and the equivalence classes of this equivalence relation are the symplectic leaves.

Example (Lie-Poisson Manifold)

If ${\mathfrak {g}}$ is a finite-dimensional Lie algebra and ${\mathfrak {g}}^{{*}}$ is its dual vector space, then the Lie bracket induces a Poisson structure on ${\mathfrak {g}}^{{*}}$ .

More precisely, we identify the cotangent bundle of the manifold ${\mathfrak {g}}^{{*}}$ , i.e., the dual of ${\mathfrak {g}}^{{*}}$ with the given Lie algebra ${\mathfrak {g}}$ . Then for two functions $f_{{1}}$ and $f_{{2}}$ on ${\mathfrak {g}}^{{*}}$ , and a point ${\mathbf {x}}\in {\mathfrak {g}}^{{*}}$ , we may define

$\{f_{1},f_{2}\}({\mathbf {x}}):=\langle \left[({\mathrm {d}}f_{{1}})_{{{\mathbf {x}}}},({\mathrm {d}}f_{{2}})_{{{\mathbf {x}}}}\right],{\mathbf {x}}\rangle$ ,

where the Lie bracket $[\cdot ,\cdot ]$ is computed in ${\mathfrak {g}}$ through the isomorphism:

${\mathrm {d}}f\in {\mathfrak {g}}^{{**}}\cong {\mathfrak {g}}$ .

If ${\mathbf {e}}_{{k}}$ are local coordinates on ${\mathfrak {g}}$ , then the Poisson bivector field is given by

${\eta _{{i,j}}}({\mathbf {x}})=\sum _{{k}}c_{{ij}}^{{k}}\langle {\mathbf {x}},{\mathbf {e}}_{{k}}\rangle$ ,

where the $c_{{ij}}^{{k}}$ are the structure constants of ${\mathfrak {g}}$ .

The symplectic leaves of this Lie-Poisson manifold are the co-adjoint orbits of the Lie algebra used for the orbit method.

Complex Structure

A complex Poisson manifold is a Poisson manifold with a complex or almost complex structure $J$ such that the complex structure preserves the bivector:

$\left(J\otimes J\right)(\eta )=\eta$ .

The symplectic leaves of a complex Poisson manifold are pseudo-Kähler manifolds.

Notes

↑ Lichnerowicz, A. (1977). "Les variétés de Poisson et leurs algèbres de Lie associées". J. Diff. Geom. 12 (2): 253–300. MR 0501133.

References

Lichnerowicz, A. (1977). "Les variétés de Poisson et leurs algèbres de Lie associées". J. Diff. Geom. 12 (2): 253–300. MR 0501133.
Kirillov, A. A. (1976). "Local Lie algebras". Russ. Math. Surv. 31 (4): 55–75. doi:10.1070/RM1976v031n04ABEH001556.
Guillemin, V.; Sternberg, S. (1984). Symplectic Techniques in Physics. New York: Cambridge Univ. Press. ISBN 0-521-24866-3.
Libermann, P.; Marle, C.-M. (1987). Symplectic geometry and analytical mechanics. Dordrecht: Reidel. ISBN 90-277-2438-5.
Bhaskara, K. H.; Viswanath, K. (1988). Poisson algebras and Poisson manifolds. Longman. ISBN 0-582-01989-3.
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. MR 834280. Errata and addenda J. Diff. Geom. 22 (1985), 255.

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