Moments and Reduction for Symplectic Groupoids
Moments and Reduction for Symplectic Groupoids
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辛群形的矩和约简
DOI：
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发表时间：
1988
期刊：
影响因子：
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通讯作者：
A. Weinstein
中科院分区：
文献类型：
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作者：
Kentaro Mikami;A. Weinstein
李群G在辛流形M上的哈密顿作用是由一个动量映射J:M-*Q*生成的，该动量映射相对于协变表示是等变的。Meyer[14]和Marsden和Weinstein[13]的约简过程是形成商M^ =J~ (//) /G^，其中/£是Q*的一个元素，G^是它的伴各向同性群。近年来（例如，参见[6]）一个已知的Lie[9]的性质被认为是必不可少的：J是一个从M到g*的具有Lie-泊松结构的泊松映射。这就提出了用任意泊松流形P3代替g*的问题，但问题立即出现了，什么对象将扮演g群的角色。这个对象刚刚被Karasev[7]和我们中的一个[3][20]确定为辛群^，本文的目的是将约简过程扩展到辛群作用。对我们工作的一个重要刺激是在b[16]中发展的泊松李群作用的约简理论。Semenov-Tian-Shansky利用Drinfel'd的泊松李群（泊松李群）的概念，解释了由完全可积系统的逆散射方法产生的修整变换的哈密顿行为。一个新的理论是必要的，因为穿衣变换不能保持它们作用的空间的泊松结构。在本文的续文中，我们希望研究泊松李群和
A hamiltonian action of a Lie group G on a symplectic manifold M is generated by a momentum map J:M-*Q* which is equivariant with respect to the coadjoint representation. The reduction procedure of Meyer [14] and Marsden and Weinstein [13] consists of forming the quotient M^ =J~ (//) /G^ where /£ is an element of Q* and G^ is its coadjoint isotropy group. In recent years (see [6], for example) a property of / already known to Lie [9] has been recognized as essential: J is a Poisson map from M to g* with its Lie-Poisson structure. This suggests the problem of replacing g* by an arbitrary Poisson manifold P3 but the question immediately arises as to what object will play the role of the group G. This object having just been identified by Karasev [7] and one of us [3] [20] as a symplectic groupoid^ the purpose of the present paper is to extend the reduction procedure to symplectic groupoid actions. An important stimulus for our work has been the reduction theory for Poisson Lie group actions developed in [16]. Using Drinfel'd's notion of Poisson Lie group [4], Semenov-Tian-Shansky explained the hamiltonian behavior of the dressing transformations which arise from the inverse-scattering approach to completely integrable systems. A new theory was necessary because dressing transformations do not preserve the Poisson structure of the spaces on which they act. In a sequel to this paper5 we hope to study how Poisson Lie groups and