nLab
Poisson manifold

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 $\{,\}:A\otimes A\to A$ such that, for any $f\in A$, $\{f,-\}:A\to A$ 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 $\{,\}:C^{\mathrm{\infty }}(M)\times C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$ 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 $C^{\mathrm{\infty }}(M)$ correspond to smooth tangent vector fields, for each $f\in C^{\mathrm{\infty }}(M)$ there is a vector $X_f$ given by $X_f(g)=\{f,g\}$ 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 $\{f,g\}:=⟨\mathrm{df}\otimes \mathrm{dg},P⟩$.

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 $f,g\in C^{\mathrm{\infty }}(N)$, $\{f\circ h,g\circ h{\}}_M=\{f,g{\}}_N$.

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.

Related concepts

• Poisson tensor

• Poisson geometry

• Poisson cohomology

• isotropic submanifold, coisotropic submanifold

• symplectic leaf

• symplectic realization

• symplectic dual pair

• Poisson Lie algebroid

• Poisson bracket Lie n-algebra

• deformation quantization

  • Moyal quantization

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

Examples of sequences of infinitesimal and local structures

first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
	$\leftarrow$ differentiation		integration $\to$
derivative		Taylor series		germ		smooth function
tangent vector		jet		germ of curve		curve
Lie algebra		formal group		local Lie group		Lie group
Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization