The modular automorphism group of a Poisson manifold
The modular automorphism group of a Poisson manifold
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DOI：
10.1016/s0393-0440(97)80011-3
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发表时间：
1997-11
影响因子：
1.5
通讯作者：
A. Weinstein
中科院分区：
文献类型：
--
作者：
A. Weinstein
Poisson流形的模流是由流形上光滑密度的选择决定的一个单参数自同构群。当密度改变时，群的生成元通过一个哈密顿向量场改变，所以一个“外自同构”的单参数群内在地附着于任何泊松流形。群是平凡的当且仅当流形容许一个在所有哈密尔顿流下不变的测度。泊松几何中的模流概念是冯诺依曼代数理论中模自同构群概念的经典极限。此外，泊松流形的模流与相关李代数胚和辛群胚的模上同调类有关。这些对象最近被证明是重要的庞加莱对偶理论的李代数胚。
The modular flow of Poisson manifold is a 1-parameter group of automorphisms determined by the choice of a smooth density on the manifold. When the density is changed, the generator of the group changes by a hamiltonian vector field, so one has a 1-parameter group of “outer automorphisms” intrinsically attached to any Poisson manifold. The group is trivial if and only if the manifold admits a measure which is invariant under all hamiltonian flows. The notion of modular flow in Poisson geometry is a classical limit of the notion of modular automorphism group in the theory of von Neumann algebras. In addition, the modular flow of a Poisson manifold is related to modular cohomology classes for associated Lie algebroids and symplectic groupoids. These objects have recently turned out to be important in Poincaré duality theory for Lie algebroids.