广义Bruhat胞腔与Kogan-Zelevinsky可积系统Hamilton流的完备性
Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems
摘要：
设$G$是任意连通和单连通复半单李群，具有标准的全纯乘法Poisson结构.证明了$G$的每一个双Bruhat胞元上的所有Fomin-Zelevinsky扭曲广义子式的Hamilton流是完备的，即它们的Hamilton向量场的所有积分曲线都定义在${\mathbb{C}}$上.由此得出$G$上的所有Kogan-Zelevinsky可积系统都有完备的Hamilton流，推广了Gekhtman和Yakimov关于$SL(n, {\mathbb{C}})$的结果.实际上，我们构造了一类具有与{ \it广义Bruhat胞元}相关联的完全Hamilton流的可积系统，这些胞元是用$G$的外尔群中任意元素序列定义的，并且我们通过（约化）双Bruhat胞元在广义Bruhat胞元中的所谓开{ \it Fomin-Zelevinsky嵌入}得到了双Bruhat胞元的结果.证明了Fomin-Zelevinsky嵌入是Poisson的，它们提供了双Bruhat胞元上的全局坐标，称为{ \it Bott-Samelson坐标}，其中所有的Fomin-Zelevinsky子式都是多项式，Poisson结构可以显式计算.
英文摘要：
Let $G$ be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of $G$ are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on ${\mathbb{C}}$. It follows that all the Kogan-Zelevinsky integrable systems on $G$ have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct a class of integrable systems with complete Hamiltonian flows associated to {\it generalized Bruhat cells} which are defined using arbitrary sequences of elements in the Weyl group of $G$, and we obtain the results for double Bruhat cells through the so-called open {\it Fomin-Zelevinsky embeddings} of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called {\it Bott-Samelson coordinates}, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.
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