{"paper":{"title":"Geometric construction of superintegrable Poisson projection chains via Poisson centralizers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The inclusions of Poisson invariants under a maximal torus and the full group form a superintegrable projection chain on semisimple Lie algebras.","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Guorui Ma, Ian Marquette, Junze Zhang, Kai Jiang, Yao-Zhong Zhang","submitted_at":"2026-05-14T07:29:32Z","abstract_excerpt":"We introduce a geometric framework for constructing superintegrable systems from Poisson centralizers (commutants) in the Lie-Poisson algebra $S(\\mathfrak{g})$ of a complex semisimple Lie algebra. Starting from a chain of reductive subgroups, we study the corresponding invariant Poisson subalgebras and their Poisson centers, and formulate superintegrability in terms of a \\emph{Poisson projection chain} of affine Poisson varieties. For a maximal torus $T\\subset G$, we prove that the inclusions $S(\\mathfrak{g})^G\\subset S(\\mathfrak{g})^T\\subset S(\\mathfrak{g})$ determine a superintegrable chain "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For a maximal torus T subset G, we prove that the inclusions S(g)^G subset S(g)^T subset S(g) determine a superintegrable chain and identify the associated quotient maps g to g//T to g//G.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The construction begins from a chain of reductive subgroups of the semisimple Lie group G, with the invariant Poisson subalgebras and their centers behaving as expected under the Poisson structure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Inclusions of invariant subalgebras S(g)^G subset S(g)^T subset S(g) for a maximal torus T in a semisimple Lie group G form a superintegrable Poisson projection chain with matching dimension splits between Hamiltonians and integrals.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The inclusions of Poisson invariants under a maximal torus and the full group form a superintegrable projection chain on semisimple Lie algebras.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a4cb6408c77a0f9c43a45bdfa7c226afd3909f650b3e51535a9d0132092f5e3c"},"source":{"id":"2605.14490","kind":"arxiv","version":1},"verdict":{"id":"8cd83901-a6dc-426c-ae33-9e2edd4e9cc9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:47:09.984239Z","strongest_claim":"For a maximal torus T subset G, we prove that the inclusions S(g)^G subset S(g)^T subset S(g) determine a superintegrable chain and identify the associated quotient maps g to g//T to g//G.","one_line_summary":"Inclusions of invariant subalgebras S(g)^G subset S(g)^T subset S(g) for a maximal torus T in a semisimple Lie group G form a superintegrable Poisson projection chain with matching dimension splits between Hamiltonians and integrals.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The construction begins from a chain of reductive subgroups of the semisimple Lie group G, with the invariant Poisson subalgebras and their centers behaving as expected under the Poisson structure.","pith_extraction_headline":"The inclusions of Poisson invariants under a maximal torus and the full group form a superintegrable projection chain on semisimple Lie algebras."},"references":{"count":41,"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":41,"snapshot_sha256":"0abee11ddd19dfd9046d04f892b4acd1893a591a41b84670a739382f53bddbfa","internal_anchors":1},"formal_canon":{"evidence_count":1,"snapshot_sha256":"2d95746ae9da9b017ce1f0043993b4c77c23c834153b566d38b1b8ae58d59c51"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}