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COMPLETENESS OF DETERMINANTAL HAMILTONIAN FLOWS ON THE MATRIX AFFINE POISSON SPACE
 

Summary: COMPLETENESS OF DETERMINANTAL HAMILTONIAN
FLOWS ON THE MATRIX AFFINE POISSON SPACE
MICHAEL GEKHTMAN AND MILEN YAKIMOV
Abstract. The matrix affine Poisson space (Mm,n, m,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 Kogan­Zelevinsky integrable
systems on Mn,n are complete and thus induce (analytic) Hamiltonian actions
of Cn(n-1)/2
on (Mn,n, n,n) (as well as on GLn(C) and on SLn(C)).
We define Gelfand­Zeitlin integrable systems on (Mn,n, n,n) from chains
of Poisson projections and prove that their flows are also complete. This is an
analog for the quadratic Poisson structure n,n of the recent result of Kostant
and Wallach [10] that the flows of the complexified classical Gelfand­Zeitlin
integrable systems are complete.
1. Introduction
The matrix affine Poisson space is the complex affine space Mm,n consisting of
rectangular matrices of size m×n equipped with the quadratic Poisson structure
(1.1) m,n =

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics