About: Poisson ring

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In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring.

Attributes	Values
rdfs:label	Poisson ring (en)
rdfs:comment	In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring. (en)
dcterms:subject	Symplectic geometry Ring theory
Wikipage page ID	4287020 (xsd:integer)
Wikipage revision ID	1124164093 (xsd:integer)
Link from a Wikipage to another Wikipage	Product rule Quantum mechanics Non-commutative algebra Algebra over a field Symplectic geometry Ring isomorphism Jacobi identity Poisson algebra Ring theory Mathematics Operator (mathematics) Classical limit Anticommutativity Commutative ring Derivation (abstract algebra) Symplectic manifold Hamiltonian mechanics Hilbert space Binary operation Symplectic geometry Distributivity Poisson bracket Symplectic form dbr:Algebra_of_smooth_functions
sameAs	Poisson ring Poisson ring Poisson ring
dbp:wikiPageUsesTemplate	dbt:PlanetMath_attribution dbt:Planetmath_reference
id	6414 (xsd:integer)
title	Poisson Ring (en) If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic (en)
urlname	IfTheAlgebraOfFunctionsOnAManifoldIsAPoissonRingThenTheManifoldIsSymplectic (en)
has abstract	In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring. Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics—the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra. (en)
prov:wasDerivedFrom	wikipedia-en:Poisson_ring?oldid=1124164093&ns=0
page length (characters) of wiki page	2434 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Poisson_ring
is Link from a Wikipage to another Wikipage of	Ring (mathematics) Poisson bracket List of things named after Siméon Denis Poisson Poisson manifold
is foaf:primaryTopic of	wikipedia-en:Poisson_ring

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