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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 a#ne 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 Kogan--Zelevinsky integrable
systems on Mn,n are complete and thus induce (analytic) Hamiltonian actions
of C n(n-1)/2 on (Mn,n , #n,n ) (as well as on GLn (C) and on SLn(C)).
We define Gelfand--Zeitlin 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 Gelfand--Zeitlin
integrable systems are complete.
1. Introduction
The matrix a#ne Poisson space is the complex a#ne space M m,n consisting of
rectangular matrices of size mn equipped with the quadratic Poisson structure
(1.1) # m,n =
m

  

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

 

Collections: Mathematics