# 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).