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

TL;DR

This paper proves the completeness of Hamiltonian flows for a broad class of integrable systems on complex semisimple Lie groups, extending previous results and constructing new systems with globally defined flows.

Contribution

It generalizes the completeness of Hamiltonian flows for Kogan-Zelevinsky systems to all double Bruhat cells and introduces integrable systems on generalized Bruhat cells using Poisson embeddings.

Findings

Hamiltonian flows are complete on all double Bruhat cells.

Constructed new integrable systems with complete flows on generalized Bruhat cells.

Fomin-Zelevinsky embeddings are Poisson and provide explicit coordinates.

Abstract

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.