Momentum Mappings And Reduction of Poisson Actions
Momentum Mappings And Reduction of Poisson Actions
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动量映射和泊松作用的约简
DOI：
10.1007/978-1-4613-9719-9_15
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
Jiang
中科院分区：
文献类型：
--
作者：
Jiang
Poisson李群G在Poisson流形P上的作用σ：G × P→P称为Poisson作用，如果σ是Poisson映射.人们认为，泊松作用应该用来理解某些可积系统的“隐藏对称性”[STS 2]。如果泊松李群G具有零泊松结构，则σ是泊松作用等价于每个变换σ g：P→ P，其中g ∈ G保持P上的泊松结构。在这种情况下，如果轨道空间G \ P是光滑流形，则它具有约化泊松结构，使得投影映射P→G \ P是泊松映射。若P是辛的，且作用量σ是由等变动量映射J：P→ g* 生成的，Meyer [Me]和Marsden和Weinstein [Ms-We]的约化过程给出了一种方法，将G \ P的辛叶描述为子P μ：= G μ \J −1（μ），其中μ∈ g*，G μ\G是μ的余伴随各向同性子群。
An action σ: G × P→P of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : P→ P for g ∈ G preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map P→G \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: P→ g*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1 (µ), where µ∈ g* and G µ ⊂ G is the coadjoint isotropy subgroup of µ.