Compact symmetric spaces, triangular factorization, and Poisson geometry

TL;DR

This paper explores the Poisson geometry of symmetric spaces using triangular decompositions, establishing links between symplectic foliations, Birkhoff decompositions, and Hamiltonian torus actions, with explicit formulas provided.

Contribution

It introduces a novel connection between Evens-Lu Poisson structures and Birkhoff decompositions on symmetric spaces, with explicit Hamiltonian torus actions and formulas.

Findings

Symplectic leaves admit natural torus actions.

The torus actions are Hamiltonian with explicitly computed momentum maps.

Local formulas for Poisson structures are derived in examples.

Abstract

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each \u-compatible triangular decomposition \g=\n_- + \h + \n_+ determines a Poisson Lie group structure \pi_U on U. The Evens-Lu construction produces a (U,\pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of \g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that the action is Hamiltonian and the momentum map is computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.