Contents

Idea

Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.

Definition

A Poisson algebra is a commutative unital associative algebra $A$, in this case over the field of real or complex numbers, equipped with a Lie bracket $\left\{,\right\}:A\otimes A\to A$ such that, for any $f\in A$, $\left\{f,-\right\}:A\to A$ is a derivation of $A$ as an associative algebra.

A Poisson manifold is a real smooth manifold $M$ equipped with a Poisson structure. A Poisson structure is a Lie algebra bracket $\left\{,\right\}:{C}^{\infty }\left(M\right)×{C}^{\infty }\left(M\right)\to {C}^{\infty }\left(M\right)$ on the vector space of smooth functions on $M$ which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of ${C}^{\infty }\left(M\right)$ correspond to smooth tangent vector fields, for each $f\in {C}^{\infty }\left(M\right)$ there is a vector ${X}_{f}$ given by ${X}_{f}\left(g\right)=\left\{f,g\right\}$ and called the Hamiltonian vector field corresponding to the function $f$, which is viewed as a classical hamiltonian function.

Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric bivector called a Poisson bivector $P\in {\Lambda }^{2}TM$; then $\left\{f,g\right\}:=⟨\mathrm{df}\otimes \mathrm{dg},P⟩$.

This induces and is equivalently encoded by the structure of a Poisson Lie algebroid.

A morphism $h:M\to N$ of Poisson manifolds is a morphism of smooth manifolds such that, for all $f,g\in {C}^{\infty }\left(N\right)$, $\left\{f\circ h,g\circ h{\right\}}_{M}=\left\{f,g{\right\}}_{N}$.

Examples

Every symplectic manifold carries a natural Poisson structure; however, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth foliations of the underlying manifold such that each leaf is a symplectic manifold.

The dual to a finite-dimensional Lie algebra has a natural structure of a Poisson manifold, the Lie-Poisson structure. Its leaves are called coadjoint orbits.

Properties

Kontsevich formality implies that every Poisson manifold has a canonical deformation quantization.

duality between algebra and geometry in physics:

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Revised on June 8, 2013 16:49:27 by Urs Schreiber (66.46.90.198)