Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space
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Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space

  • Author(s): Gekhtman, Michael
  • Yakimov, Milen
  • et al.
Abstract

The matrix affine Poisson space (M m,n , π 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 $${{\rm GL}_n(\mathbb{C})}$$ . We prove that the Hamiltonian flows of all minors are complete. As a corollary we obtain that all Kogan–Zelevinsky integrable systems on M n,n are complete and thus induce (analytic) Hamiltonian actions of $${\mathbb{C}^{n(n-1)/2}}$$ on (M n,n , π n,n ) (as well as on $${{\rm GL}_n(\mathbb{C})}$$ and on $${{\rm SL}_n(\mathbb{C})}$$ ). We define Gelfand–Zeitlin integrable systems on (M n,n , π n,n ) from chains of Poisson projections and prove that their flows are also complete. This is an analog for the quadratic Poisson structure π n,n of the recent result of Kostant and Wallach (Studies in Lie Theory. Progress in Mathematics, vol 243, pp 319–364. Birkhäuser, Boston, 2006) that the flows of the complexified classical Gelfand–Zeitlin integrable systems are complete.

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