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Let $G$ be any connected and simply connected complex semisimple Lie group,equipped with a standard holomorphic multiplicative Poisson structure. We showthat the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalizedminors on every double Bruhat cell of $G$ are complete in the sense that allthe 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 andYakimov for the case of $SL(n, {\mathbb{C}})$. We in fact construct a class ofintegrable systems with complete Hamiltonian flows associated to {\itgeneralized Bruhat cells} which are defined using arbitrary sequences ofelements in the Weyl group of $G$, and we obtain the results for double Bruhatcells through the so-called open {\it Fomin-Zelevinsky embeddings} of (reduced)double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinskyembeddings are proved to be Poisson, and they provide global coordinates ondouble Bruhat cells, called {\it Bott-Samelson coordinates}, in which all theFomin-Zelevinsky minors become polynomials and the Poisson structure can becomputed explicitly.

Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems