Collective superintegrable systems from the Guillemin--Sternberg torus action

2026/08/03 by L. Feher
Physics and Astronomy · Mathematics · #math-ph #math.MP

paper · pdf

12 pages

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, G, on a symplectic manifold, M. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of \mathfrakH := J^*(C^∞(\mathfrakg^*)^G), where J: M → \mathfrakg^* is the momentum map of the G action, and its centralizer \mathfrakF in C^∞(M) satisfy the equality ddim(\mathfrakH) + ddim(\mathfrakF) = dim(M). Together with a non-triviality condition, this ensures that the Abelian Poisson algebra \mathfrakH⊂ C^∞(M) represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.

Citations

• Integrable systems from Poisson reductions of generalized Hamiltonian torus actions
• Degenerately Integrable Systems
• Torus Actions and Integrable Systems
• Non-commutative Integrability, Moment Map and Geodesic Flows
• Non-abelian convexity by symplectic cuts
• Non-abelian convexity by symplectic cuts
• Multiplicity-free Hamiltonian actions need not be Kähler