ii Multiplicative and Affine Poisson Structures on Lie Groups
ii Multiplicative and Affine Poisson Structures on Lie Groups
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发表时间：
2011
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通讯作者：
Jiang-Hua Lu;A. Weinstein
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作者：
Jiang-Hua Lu;A. Weinstein
本文研究了李群的两类特殊Poisson结构，即乘法Poisson结构和仿射Poisson结构。具有乘法泊松结构的李群称为泊松李群。首先回顾Drinfel'd和Semenov-Tian-Shansky已知的Poisson李群的基本性质，我们发展了Poisson李群及其动量映射的Poisson作用理论。我们将展示如何进行通常的Marsden-Weinstein辛减少在这个一般的设置。给出了Poisson李群的辛群胚的一个构造。仿射泊松结构比乘法结构更一般，但它们也有非常简单的性质。例如，它已经知道达佐德和Sondaz，他们的辛叶是轨道的“敷料行动”。证明了Dressing作用是Poisson作用，其Poisson上同调可以用对偶李代数的李代数上同调来实现，并且可以系统地构造其辛群胚.另一方面，李群G上的仿射Poisson结构定义了G上乘法Poisson结构之间的等价关系。它们也可以作为（不一定是等变的）动量映射的目标泊松结构。
In this dissertation, we study two special classes of Poisson structures of Lie groups, namely multiplicative and affine Poisson structures. A Lie group equipped with a multiplicative Poisson structure is called a Poisson Lie group. Starting with a review of the basic properties of Poisson Lie groups already known to Drinfel’d and Semenov-Tian-Shansky, we develop the theory of Poisson actions by Poisson Lie groups and their momentum mappings. We show how to carry out the usual Marsden-Weinstein symplectic reduction in this general setting. A construction of symplectic groupoids for Poisson Lie groups is given. Affine Poisson structures are more general then multiplicative ones, but they also have very simple properties. For example, it is already known to Dazord and Sondaz that their symplectic leaves are orbits of the “dressing actions”. We show that the dressing actions are Poisson actions; their Poisson cohomology can be realized as Lie algebra cohomology of their dual Lie algebras, and their symplectic groupoids can be constructed in a systematical way. On the other hand, affine Poisson structures on a given Lie group G define an equivalence relation between multiplicative Poisson structures on G. They also serve as the target Poisson structures for (not necessarily equivariant) momentum mappings.