{"paper":{"title":"Integrable systems from Poisson reductions of generalized Hamiltonian torus actions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Sufficient conditions let an integrable system with symmetry K descend to an integrable system on the dense open set of the quotient Poisson space M/K.","cross_cats":["hep-th","math.MP","math.SG","nlin.SI"],"primary_cat":"math-ph","authors_text":"L. Feher, M. Fairon","submitted_at":"2025-07-16T09:11:05Z","abstract_excerpt":"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 reduc"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The unreduced system on M is supposed to possess action variables that generate a proper, effective action of a group of the form U(1)^ℓ1 × R^ℓ2 and descend to action variables of the reduced system (abstract, paragraph on generalized Hamiltonian torus actions).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Develops sufficient conditions for Poisson reduction of generalized Hamiltonian torus actions to preserve integrability and applies them to open problems on Lie group doubles and flat-connection moduli spaces.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Sufficient conditions let an integrable system with symmetry K descend to an integrable system on the dense open set of the quotient Poisson space M/K.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a34eddae5c97dbe3e8f642c1982f2a59bbbd01593c58fc3c5a965ded8ee22827"},"source":{"id":"2507.12051","kind":"arxiv","version":2},"verdict":{"id":"74732f9b-7262-43e6-b0e4-16ab871ee4e4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T04:49:35.898427Z","strongest_claim":"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.","one_line_summary":"Develops sufficient conditions for Poisson reduction of generalized Hamiltonian torus actions to preserve integrability and applies them to open problems on Lie group doubles and flat-connection moduli spaces.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The unreduced system on M is supposed to possess action variables that generate a proper, effective action of a group of the form U(1)^ℓ1 × R^ℓ2 and descend to action variables of the reduced system (abstract, paragraph on generalized Hamiltonian torus actions).","pith_extraction_headline":"Sufficient conditions let an integrable system with symmetry K descend to an integrable system on the dense open set of the quotient Poisson space M/K."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.12051/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"74757d72504b45d948534956b8f06d745ea6927d042262882c32d3a7c480ba9c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}