Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space
نویسندگان
چکیده
The matrix affine Poisson space (Mm,n, πm,n) is the space of complex rectangular matrices equipped with a canonical quadratic Poisson structure which in the square case m = n reduces to the standard Poisson structure on GLn(C). We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan–Zelevinsky integrable systems on Mn,n are complete and thus induce (analytic) Hamiltonian actions of C on (Mn,n, πn,n) (as well as on GLn(C) and on SLn(C)). We define Gelfand–Zeitlin integrable systems on (Mn,n, πn,n) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure πn,n of the recent result of Kostant and Wallach [10] that the flows of the complexified classical Gelfand–Zeitlin integrable systems are complete.
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