University of Toronto's Symplectic Geometry Seminar

Monday January 31, 2005, 14:10--15:00
SS 5017A

Eckhard Meinrenken

U of Toronto

Poisson Lie groups and the Gelfand-Zeitlin system

Abstract: The Gelfand-Zeitlin parameters of a Hermitian n x n matrix A are the n(n+1)/2 numbers given as eigenvalues of all its `upper left corner' k x k submatrices. We show that there is a canonical diffeomorphism f from Hermitian matrices onto positive definite matrices, such that the Gelfand-Zeitlin parameters of f(A) are the exponentials of those of A. The construction of this map involves techniques from Poisson geometry, and depends in particular on the proof of a conjecture of Flaschka and Ratiu about a Poisson-Lie group version of the Gelfand-Zeitlin system.

Joint work with Anton Alekseev (Geneva).