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Collective superintegrable systems from the Guillemin--Sternberg torus action

  • L. Feher
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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 H := 𝒥*(C∞(g*)G), where 𝒥: M → g* is the momentum map of the G action, and its centralizer F in C∞(M) satisfy the equality ddim(H) + ddim(F) = dim(M). Together with a non-triviality condition, this ensures that the Abelian Poisson algebra H⊂ 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.

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Cite

BibTeXRIS
BibTeX
@misc{feher2026collective,
  title = {{Collective superintegrable systems from the Guillemin--Sternberg torus action}},
  author = {L. Feher},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.01878},
  eprint = {2608.01878},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.01878},
}
RIS
TY  - UNPB
TI  - Collective superintegrable systems from the Guillemin--Sternberg torus action
AU  - L. Feher
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.01878
UR  - https://arxiv.org/abs/2608.01878
AB  - 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 H := 𝒥^*(C^∞(g^*)^G), where 𝒥: M → g^* is the momentum map of the G action, and its centralizer F in C^∞(M) satisfy the equality ddim(H) + ddim(F) = dim(M). Together with a non-triviality condition, this ensures that the Abelian Poisson algebra H⊂ 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.
ER  -