Integrable systems from Poisson reductions of generalized Hamiltonian torus actions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917512728281088 |
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| author | Feher, L. Fairon, M. |
| author_facet | Feher, L. Fairon, M. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrable systems from Poisson reductions of generalized Hamiltonian torus actions Feher, L. Fairon, M. Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems 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. |
| title | Integrable systems from Poisson reductions of generalized Hamiltonian torus actions |
| topic | Mathematical Physics High Energy Physics - Theory Symplectic Geometry Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2507.12051 |