Homogeneous Poisson structures on symmetric spaces

TL;DR

This paper explicitly computes Hamiltonian systems from homogeneous Poisson structures on symmetric spaces, revealing isomorphisms between systems in compact and noncompact cases and connections to Bruhat Poisson structures.

Contribution

It provides explicit calculations of Hamiltonian systems on symmetric spaces and demonstrates their isomorphisms, linking different Poisson structures and factoring them via known results.

Findings

Hamiltonian systems are explicitly calculated for symmetric spaces.

Systems in noncompact and compact cases are isomorphic.

Connections established with Bruhat Poisson structures and their factorization.

Abstract

We calculate, in a relatively explicit way, the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. A corollary is that the Hamiltonian system arising in the noncompact case is isomorphic to the generic Hamiltonian system arising in the compact case. In the group case these systems are also isomorphic to those arising from the Bruhat Poisson structure on the flag space, and hence, by results of Lu, can be completely factored.