Generalized Bruhat Cells and Completeness of the Kogan-Zelevinsky Integrable Systems. (arXiv:1708.01375v1 [math.RT])

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 of their integral curves are defined on ${\mathbb{C}}$. It follows that the Kogan-Zelevinsky integrable systems on $G$ are all complete, generalizing the result of Gekhtman and Yakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct a class of complete Hamiltonian flows and complete integral systems related to any {\it generalized Bruhat cell} which is defined using an arbitrary sequence 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 to generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide glob 查看全文>>