一类3维广义雅可比矩阵Poisson结构的同调性质
Homological properties of certain generalized Jacobian Poisson structures in dimension 3
摘要：
泊松结构的单模性条件（即，具有平凡模类的Poisson结构）在其Poisson同调和Poisson上同调之间导出了Poincaré对偶。因此，这类Poisson结构的Poisson同调信息通过对偶导出其Poisson上同调信息，反之亦然。但在非平凡模块类的情况下，这不再是正确的。这就是广义雅可比矩阵泊松结构（GJPS）的情况。本文考虑一类3维GJPS，得到了它们的Poisson同调群和Poisson上同调群的性质.更确切地说，在一些假设下，我们得到了这些泊松同调群的庞加莱级数，我们明确计算这些泊松上同调群，除了第二组似乎更复杂的获得。
英文摘要：
The unimodularity condition for a Poisson structure (ie., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. Therefore an information about the Poisson homology of this kind Poisson structures induces by duality an information about its Poisson cohomology and vise versa. But this is not longer true in the case of a non trivial modular class. That is the case of Generalized Jacobian Poisson Structures (GJPS). In this paper, we consider certain GJPS in dimension 3 and obtain properties of their Poisson homological groups and their Poisson cohomological groups. More precisely, under some assumptions, we obtain the Poincaré series of these Poisson homological groups and we compute explicitly these Poisson cohomological groups, except the second group which seems more complicated to obtain.
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