Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

TL;DR

This paper establishes conditions under which integrable systems with symmetries can be reduced to integrable systems on quotient Poisson spaces, with applications to various geometric and algebraic structures.

Contribution

It introduces a framework for reducing integrable systems with generalized Hamiltonian torus actions, including quasi-Poisson cases, and applies it to solve open problems in Lie group reductions.

Findings

Provided sufficient conditions for integrability descent on quotient spaces

Applied the framework to systems on doubles of compact Lie groups

Connected the reduction process to moduli spaces of flat connections

Abstract

We develop a set of sufficient conditions for guaranteeing that an integrable system with a symmetry group $K$ on a manifold $M$ descends to an integrable system on a dense open subset of the quotient Poisson space $M/K$. The higher dimensional phase space $M$ carries a bivector $P_M$ yielding a bracket on $C^\infty(M)$ such that $C^\infty(M)^K$ is a Poisson algebra. The unreduced system on $M$ is supposed to possess `action variables' that generate a proper, effective action of a group of the form $\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$ and descend to action variables of the reduced system. In view of the form of the group and since $P_M$ could be a quasi-Poisson bivector, we say that we work with a generalized Hamiltonian torus action. The reduced systems are in general superintegrable owing to the large set of invariants of the proper Hamiltonian action of $\mathrm{U}(1)^{\ell_1} \times \mathbb{R}^{\ell_2}$. We present several examples and apply our construction for solving open problems regarding the integrability of systems obtained previously by reductions of master systems on doubles of compact Lie groups: the cotangent bundle, the Heisenberg double and the quasi-Poisson double. Furthermore, we offer numerous applications to integrable systems living on moduli spaces of flat connections, using the quasi-Poisson approach.