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topology > algebraic topology > manifold > differentiable manifold > Poisson manifold

geometry > topological space > manifold > differentiable manifold > Poisson manifold

geometry > differential geometry > differentiable manifold > Poisson manifold

topology > differential topology > differentiable manifold > Poisson manifold

Preferred term

Poisson manifold

Definition(s)

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
A Poisson structure (or Poisson bracket) on a smooth manifold ${\\displaystyle M}$ is a function ${\\displaystyle \\{\\cdot ,\\cdot \\}:{\\mathcal {C}}^{\\infty }(M)\ imes {\\mathcal {C}}^{\\infty }(M)\ o {\\mathcal {C}}^{\\infty }(M)}$ on the vector space ${\\displaystyle {C^{\\infty }}(M)}$ of smooth functions on ${\\displaystyle M}$ , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Poisson_manifold)

Broader concept(s)

differentiable manifold

Narrower concept(s)

Poisson-Lie group

In other languages

variété de Poisson

French

URI

http://data.loterre.fr/ark:/67375/PSR-KZLFZ6JQ-N

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