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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Acknowledgements
We would like to thank Yvette Kosmann-Schwarzbach and the referees for many valuable comments which helped us to improve the paper a lot. M.Y. would like to thank the organizers of the conference Poisson 2008 for the invitation to participate in this very stimulating conference. The research of M. G. was partially supported by NSF grant DMS-0801204. The research of M.Y. was partially supported by NSF grant DMS-0701107 and an Alfred P. Sloan Research Fellowship.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Gekhtman, M., Yakimov, M. Completeness of Determinantal Hamiltonian Flows on the Matrix Affine Poisson Space. Lett Math Phys 90, 161–173 (2009). https://doi.org/10.1007/s11005-009-0337-0
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DOI: https://doi.org/10.1007/s11005-009-0337-0