Ginzburg-weinstein via gelfand-zeitlin

A. Alekseev University of Geneva E. Meinrenken University of Toronto

Differential Geometry mathscidoc:1609.10015

Journal of Differential Geometry, 76, (1), 1-34, 2007
Let U(n) be the unitary group, and u(n)¤ the dual of its Lie algebra, equipped with the Kirillov Poisson structure. In their 1983 paper, Guillemin-Sternberg introduced a densely defined Hamiltonian action of a torus of dimension (n−1)n/2 on u(n)¤, with moment map given by the Gelfand-Zeitlin coordinates. A few years later, Flaschka-Ratiu described a similar, ‘multiplicative’ GelfandZeitlin system for the Poisson Lie group U(n)¤. By the Ginzburg-Weinstein theorem, U(n)¤ is isomorphic to u(n)¤ as a Poisson manifold. Flaschka-Ratiu conjectured that one can choose the Ginzburg-Weinstein diffeomorphism in such a way that it intertwines the linear and nonlinear Gelfand-Zeitlin systems. Our main result gives a proof of this conjecture, and produces a canonical Ginzburg-Weinstein diffeomorphism.
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  author={A. Alekseev, and E. Meinrenken},
  booktitle={Journal of Differential Geometry},
A. Alekseev, and E. Meinrenken. GINZBURG-WEINSTEIN VIA GELFAND-ZEITLIN. 2007. Vol. 76. In Journal of Differential Geometry. pp.1-34.
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