Poisson manifold (original) (raw)
Als Poisson-Mannigfaltigkeit bezeichnet man in der Mathematik eine differenzierbare Mannigfaltigkeit, die mit einer Poisson-Struktur versehen ist. Eine Poisson-Struktur ist eine bilineare Abbildung auf der Algebra der glatten Funktionen, welche die Eigenschaften einer Poisson-Klammer erfüllt. Benannt sind die Poisson-Mannigfaltigkeit, -Struktur und -Klammer nach dem Physiker und Mathematiker Siméon Denis Poisson.
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| dbo:abstract | Als Poisson-Mannigfaltigkeit bezeichnet man in der Mathematik eine differenzierbare Mannigfaltigkeit, die mit einer Poisson-Struktur versehen ist. Eine Poisson-Struktur ist eine bilineare Abbildung auf der Algebra der glatten Funktionen, welche die Eigenschaften einer Poisson-Klammer erfüllt. Benannt sind die Poisson-Mannigfaltigkeit, -Struktur und -Klammer nach dem Physiker und Mathematiker Siméon Denis Poisson. (de) En géométrie, une structure de Poisson sur une variété différentielle est un crochet de Lie (appelé crochet de Poisson dans ce cas) sur l'algèbre des fonctions lisses de à valeurs réelles, vérifiant formule de Leibniz . En d'autres termes, une structure de Poisson est structure d'algèbre de Lie sur l'espace vectoriel des fonctions lisses sur de sorte que est un champ de vecteurs pour toute fonction lisse , appelé champ de vecteurs hamiltonien associé à . (fr) En geometría simpléctica, una variedad de Poisson es una variedad diferenciable provista de un paréntesis de Lie (el paréntesis de Poisson es un caso especial) definido sobre el álgebra de funciones suaves sobre , que satisface la regla de Leibniz: Es decir, se trata de un estructura de álgebra de Lie definida sobre el espacio vectorial de funciones suaves sobre tal que es un campo vectorial para cada función suave , que denominamos asociado a . Estos campos vectoriales generan una foliación singular completamente integrable, heredadando cada una de sus subvariedades integrales maximales una estructura simpléctica. Uno podría, por tanto, afirmar informalmente que una variedad de Poisson admite una partición suave de en hojas simplécticas de dimensión par, aunque no todas ellas tienen por qué tener la misma dimensión.Las variedade de Poisson son un caso particular de estructuras de Jacobi introducidas por André Lichnerowicz en 1977. Estas variedades fueron clasificadas en un artículo clásico de , donde muchos teoremas sobre la estructura básica fueron demostrados por primera vez y que ejerció una enorme influencia en el desarrollo de la geometría de Poisson, que actualmente está profundamente relacionado con la geometría no conmutativa, los sistemas integrables, las y la teoría de la representación, por nombrar algunos campos con los que se han establecido relaciones. (es) In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule . Equivalently, defines a Lie algebra structure on the vector space of smooth functions on such that is a vector field for each smooth function (making into a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977. They were further studied in the classical paper of Alan Weinstein, where many basic structure theorems were first proved, and which exerted a huge influence on the development of Poisson geometry — which today is deeply entangled with non-commutative geometry, integrable systems, topological field theories and representation theory, to name a few. Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics. (en) 多様体 M がポアソン多様体（ポアソンたようたい、英: Poisson Manifold）であるとは、M 上の C∞ 級関数全体のなすベクトル空間を C∞(M) と表すとき、次の性質を満たす写像 が存在することをいう。 1. * は、-双線形形式である。 2. * 3. * :ヤコビ律 4. * このとき、写像 を M 上のポアソン構造、もしくはポアソン括弧と呼ぶ。 (ja) Una varietà di Poisson è una varietà differenziabile dotata di una struttura aggiuntiva che generalizza quella presente nelle varietà simplettiche e quindi anche la struttura simplettica canonica di un fibrato cotangente tramite cui si formalizza la meccanica hamiltoniana. Le varietà di Poisson sono caratterizzate dalla presenza di un'operazione sulle funzioni che soddisfa le proprietà essenziali delle classiche parentesi di Poisson definite su un fibrato cotangente; tramite questa operazione è possibile associare ad ogni funzione un campo hamiltoniano che generalizza le usuali equazioni di Hamilton. Molte delle definizioni e dei risultati del formalismo hamiltoniano possono essere estese a questo contesto più ampio. (it) 미분기하학에서 푸아송 다양체(Poisson多樣體, 영어: Poisson manifold)는 푸아송 괄호를 정의할 수 있는 심플렉틱 다양체의 일반화이다. 심플렉틱 다양체와 달리 괄호가 일부 점에서 퇴화할 수 있다. (ko) 在数学中，泊松流形（Poisson manifold）是一个微分流形 M 使得 M 上光滑函数代数 C∞(M) 上装备有一个双线性映射称为泊松括号，将其变成泊松代数。 每个辛流形是泊松流形，反之则不然。 (zh) |
| dbo:wikiPageExternalLink | https://archive.org/details/symplecticgeomet0000libe https://www.ams.org/bull/1996-33-02/S0273-0979-96-00644-1/S0273-0979-96-00644-1.pdf https://bookstore.ams.org/gsm-217 https://faculty.math.illinois.edu/~ruiloja/Math595/Spring14/book.pdf |
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| rdfs:comment | Als Poisson-Mannigfaltigkeit bezeichnet man in der Mathematik eine differenzierbare Mannigfaltigkeit, die mit einer Poisson-Struktur versehen ist. Eine Poisson-Struktur ist eine bilineare Abbildung auf der Algebra der glatten Funktionen, welche die Eigenschaften einer Poisson-Klammer erfüllt. Benannt sind die Poisson-Mannigfaltigkeit, -Struktur und -Klammer nach dem Physiker und Mathematiker Siméon Denis Poisson. (de) En géométrie, une structure de Poisson sur une variété différentielle est un crochet de Lie (appelé crochet de Poisson dans ce cas) sur l'algèbre des fonctions lisses de à valeurs réelles, vérifiant formule de Leibniz . En d'autres termes, une structure de Poisson est structure d'algèbre de Lie sur l'espace vectoriel des fonctions lisses sur de sorte que est un champ de vecteurs pour toute fonction lisse , appelé champ de vecteurs hamiltonien associé à . (fr) 多様体 M がポアソン多様体（ポアソンたようたい、英: Poisson Manifold）であるとは、M 上の C∞ 級関数全体のなすベクトル空間を C∞(M) と表すとき、次の性質を満たす写像 が存在することをいう。 1. * は、-双線形形式である。 2. * 3. * :ヤコビ律 4. * このとき、写像 を M 上のポアソン構造、もしくはポアソン括弧と呼ぶ。 (ja) 미분기하학에서 푸아송 다양체(Poisson多樣體, 영어: Poisson manifold)는 푸아송 괄호를 정의할 수 있는 심플렉틱 다양체의 일반화이다. 심플렉틱 다양체와 달리 괄호가 일부 점에서 퇴화할 수 있다. (ko) 在数学中，泊松流形（Poisson manifold）是一个微分流形 M 使得 M 上光滑函数代数 C∞(M) 上装备有一个双线性映射称为泊松括号，将其变成泊松代数。 每个辛流形是泊松流形，反之则不然。 (zh) En geometría simpléctica, una variedad de Poisson es una variedad diferenciable provista de un paréntesis de Lie (el paréntesis de Poisson es un caso especial) definido sobre el álgebra de funciones suaves sobre , que satisface la regla de Leibniz: (es) In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule . Equivalently, defines a Lie algebra structure on the vector space of smooth functions on such that is a vector field for each smooth function (making into a Poisson algebra). Poisson structures are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics. (en) Una varietà di Poisson è una varietà differenziabile dotata di una struttura aggiuntiva che generalizza quella presente nelle varietà simplettiche e quindi anche la struttura simplettica canonica di un fibrato cotangente tramite cui si formalizza la meccanica hamiltoniana. (it) |
| rdfs:label | Poisson-Mannigfaltigkeit (de) Variedades de Poisson (es) Variété de Poisson (fr) Varietà di Poisson (it) 푸아송 다양체 (ko) ポアソン多様体 (ja) Poisson manifold (en) 泊松流形 (zh) |
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