nLab Poisson algebra





A Poisson algebra is

  • a module AA over some field or other commutative ring kk,

  • equipped with the structure :A kAA{\cdot}\colon A \otimes_k A \to A of a commutative associative algebra;

  • and equipped with the bracket [,]:A kAA[-,-]\colon A \otimes_k A \to A of a Lie algebra;

  • such that for every aAa \in A we have that [a,]:AA[a,-]\colon A \to A is a derivation of (A,)(A,\cdot).

The definition makes sense, but is not standardly used, also for the more general case when the product \cdot is not necessarily commutative (it is often however taken in the sense of commutative internally to a symmetric monoidal category, say of chain complexes, graded vector spaces or supervector spaces).

Poisson algebras form a category Poiss.


The opposite category of that of (commutative) real Poisson algebras can be identified with the category of classical mechanical systems

ClassMechSys:=CPoiss op. ClassMechSys := CPoiss^{op} \,.

See there for more details.


For (X,{,})(X, \{-,-\}) a Poisson manifold or (X,ω)(X, \omega) a symplectic manifold, the algebra of smooth functions C (X,)C^\infty(X, \mathbb{R}) is naturally a Poisson algebra, thus may be regarded as an object in ClassMechSysClassMechSys. For classical mechanical systems of this form, we say that the manifold XX is the phase space of the system.

Generally, therefore, for (A,,[,])(A, \cdot,[-,-]) a Poisson algebra, we may regard it as a formal dual to some generalized Phase space.


For (A,,{,})(A, \cdot, \{-,-\}) a Poisson algebra, AA together with its module Ω 1(A)\Omega^1(A) of Kähler differentials naturally form a Lie-Rinehart pair, with bracket given by

[da,db]:=d{a,b}. [d a, d b ] := d \{a,b\} \,.

If the Poisson algebra comes from a Poisson manifold XX, then this Lie-Rinehart pair is the Chevalley-Eilenberg algebra of the given Poisson Lie algebroid over XX. We can therefore identify classical mechanical systems over a phase space manifold also with Poisson Lie algebroids.


For a symplectic manifold


A symplectic manifold (X,ω)(X, \omega) canonically is a Poisson manifold (X;{,})(X; \{-,-\}) by defining the Poisson bracket as follows.

By the symplectic structure, to every smooth function fC (X)f \in C^\infty(X) is associated the correspinding Hamiltonian vector field v fΓ(TX)v_f \in \Gamma(T X), defined, uniquely, by the equation

d dRf=ι v fω. d_{dR} f = \iota_{v_f} \omega \,.

In terms of this, the Poisson bracket is given by

{f,g}:=ι v gι v fω. \{f,g\} := \iota_{v_g} \iota_{v_f} \omega \,.


For a symplectic manifold

Let (X,ω)(X, \omega) be a symplectic manifold, and (X,{,})(X, \{-,-\}) the corresponding Poisson manifold as above.

Write 𝒫:=(C (X),{,})\mathcal{P} := (C^\infty(X), \{-,-\}) for the Lie algebra underlying the Poisson algebra.


This fits into a central extension of Lie algebras

𝒫Ham(X), \mathbb{R} \to \mathcal{P} \to Ham(X) \,,

where Ham(X)Γ(TX)Ham(X) \subset \Gamma(T X) is the sub-Lie algebra of vector fields on the Hamiltonian vector fields.


Observe that the Hamiltonian function associated to a Hamiltonian vector field is well-defined only up to addition of a constant function.

This is also called the Kostant-Souriau central extension (see Kostant 1970).

duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}


Discussion of Lie integration of the Poisson brackets on non-compact manifolds, by restriction to compactly supported functions:

  • Augustin Banyaga, Paul Donato, Some remarks on the integration of the Poisson algebra, Journal of Geometry and Physics 19 4 (1996) 368-378 [doi:10.1016/0393-0440(95)00039-9]

Last revised on July 3, 2023 at 14:47:24. See the history of this page for a list of all contributions to it.