2017/08/04 by Jiang-Hua Lu, Lu, Jiang-Hua, Yipeng Mi +1
Mathematics · #17B80 #22E46 #53D17 #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.RT #math.SG #msc:17B80 #msc:22E46 #msc:53D17
paper · pdf · doi:10.48550/arxiv.1708.01375
Title slightly changed; Section 1.3 expanded; some typos corrected
arxiv created 2017/11/01 · arxiv updated 2017/11/02
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 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, 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.