{"paper":{"title":"Poisson Centralisers and Polynomial Superintegrability for Magnetic Geodesic Flows on Reductive Homogeneous Spaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Magnetic geodesic flows on reductive homogeneous spaces admit superintegrability through polynomial integrals from the Lie algebra and an invariant slice.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Guorui Ma, Ian Marquette, Junze Zhang, Kai Jiang, Yao-Zhong Zhang","submitted_at":"2026-01-04T04:41:28Z","abstract_excerpt":"We provide a method for formulating superintegrable magnetic geodesic flows on reductive homogeneous spaces $M=G/A$, with $G$ a compact semisimple Lie group and $A$ a closed subgroup of $G$. In the twisted cotangent bundle $(T^*M,\\omega_\\varepsilon)$, with $\\omega_\\varepsilon=\\omega_{\\mathrm{can}}+\\varepsilon\\,\\pi^*\\omega_{\\mathrm{KKS}}$ being the canonical plus Kirillov-Kostant-Souriau (KKS) forms, we build two canonical and commuting families of polynomial first integrals: one pulled back from the Lie algebra $\\mathfrak{g}$ of $G$ via the magnetic moment map $P$, and one pulled back from a $"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Their common image generates a reduced Poisson algebra obtained from a fiber tensor product, and the natural multiplication map into a Poisson subalgebra of polynomial functions O(T^*M) subset C^infty(T^*M) is Poisson and injective. In a dense regular locus, the resulting projection chain realises a superintegrable system.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The construction assumes the existence of an Ad(A)-invariant affine slice of m congruent to T_{eA}M and that the magnetic moment map P pulls back commuting polynomial integrals from g; this is stated as part of the setup for reductive homogeneous spaces but not derived in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Constructs two commuting families of polynomial first integrals for magnetic geodesic flows on reductive homogeneous spaces G/A, yielding a superintegrable system via a reduced Poisson algebra in a dense regular locus.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Magnetic geodesic flows on reductive homogeneous spaces admit superintegrability through polynomial integrals from the Lie algebra and an invariant slice.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"75ae7306df0050970e0096045d828f00d367044c9ca350ad8c73fb291f15d198"},"source":{"id":"2601.01369","kind":"arxiv","version":2},"verdict":{"id":"0652306f-7d56-4523-a652-bff6a6f40a23","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T18:21:02.858470Z","strongest_claim":"Their common image generates a reduced Poisson algebra obtained from a fiber tensor product, and the natural multiplication map into a Poisson subalgebra of polynomial functions O(T^*M) subset C^infty(T^*M) is Poisson and injective. In a dense regular locus, the resulting projection chain realises a superintegrable system.","one_line_summary":"Constructs two commuting families of polynomial first integrals for magnetic geodesic flows on reductive homogeneous spaces G/A, yielding a superintegrable system via a reduced Poisson algebra in a dense regular locus.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The construction assumes the existence of an Ad(A)-invariant affine slice of m congruent to T_{eA}M and that the magnetic moment map P pulls back commuting polynomial integrals from g; this is stated as part of the setup for reductive homogeneous spaces but not derived in the abstract.","pith_extraction_headline":"Magnetic geodesic flows on reductive homogeneous spaces admit superintegrability through polynomial integrals from the Lie algebra and an invariant slice."},"references":{"count":44,"sample":[{"doi":"","year":2013,"title":"W. Miller Jr, S. Post, and P. Winternitz. Classical and quantum superintegrability with applications.J. Phys. A: Math. Theor., 46(42):423001, 97, 2013","work_id":"a6a5522e-2097-400e-9e22-e9137a4f4a64","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"N. Reshetikhin. Degenerately integrable systems.Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 433(Voprosy Kvantovo˘ ı Teorii Polya i Statistichesko˘ ı Fiziki. 23):224–245, 2015","work_id":"3c1bab89-d9a2-4644-b368-7a7e866faa88","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1965,"title":"J. Friˇ s, V. Mandrosov, Ya. A. Smorodinsky, M. Uhl´ ıˇ r, and P. Winternitz. On higher symmetries in quantum mechanics.Phys. Lett., 16:354–356, 1965","work_id":"40834b6e-408c-4920-bf10-4602a34dee49","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1968,"title":"Action-angle variables and their generalizations.Trans","work_id":"11659639-2aa5-4b47-b1d2-8c2fc6948174","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"E. G. Kalnins, J. M. Kress, and W. Miller, Jr.Separation of variables and superintegrability. IOP Expanding Physics. IOP Publishing, Bristol, 2018. The symmetry of solvable systems","work_id":"4038fbf4-774b-428d-8ad8-f0c97f4af1f8","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"bc67427548081d601d3418be3739bd748b120f5241e99600281c4caa0c9aaa66","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ab1639b888866561f8c634a186e11ec7836d8daa15de2696994acad2b26dfa7a"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}