Poisson流形的群轨道与正则分拆
Group orbits and regular partitions of Poisson manifolds
摘要：
我们研究了一个大类的泊松流形，来自Manin三元组，为此我们构造明确的分区成定期泊松子流形相交某些组轨道。例子包括还原二次李代数$\d$的拉格朗日子代数的所有变种${\mathcal L}$，其泊松结构由$\d$的拉格朗日分裂定义。在$\g \oplus \g$的特殊情况下，其中$\g$是一个复杂的半单李代数，我们显式计算的${\mathcal L}$上的泊松结构的秩定义的任意拉格朗日分裂的${\mathfrak g} \oplus {\mathfrak g}$。这种拉格朗日分裂已被P. Delorme分类，它们包含Belavin-Drinfeld分裂作为特殊情况。
英文摘要：
We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties ${\mathcal L}$ of Lagrangian subalgebras of reductive quadratic Lie algebras $\d$ with Poisson structures defined by Lagrangian splittings of $\d$. In the special case of $\g \oplus \g$, where $\g$ is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on ${\mathcal L}$ defined by arbitrary Lagrangian splittings of ${\mathfrak g} \oplus {\mathfrak g}$. Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin--Drinfeld splittings as special cases.
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