Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

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.

45 pages, 1 figure, v2: 46 pages, with a few remarks and 2 references added, and typos corrected, v3: a few more typos removed, final version