Noncommutative Integrability, Moment Map and Geodesic Flows

Abstract

The purpose of this paper is to discuss the relationship between commutative and noncommutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in the noncommutative sense by means of polynomial integrals, and therefore, in the classical commutative sense by means of C∞-smooth integrals.