{"paper":{"title":"Quasi-Poisson Modules and Harish-Chandra $\\bs{AD}$-Modules","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules.","cross_cats":[],"primary_cat":"math.RT","authors_text":"Malihe Yousofzadeh","submitted_at":"2026-05-16T12:02:34Z","abstract_excerpt":"We introduce the notion of quasi-Poisson modules over Lie-Rinehart pairs and prove that for the Lie-Rinehart pair $(\\dot A,\\dot\\fk)$ in which $\\dot A=\\bbbc[t_1^{\\pm1},\\ldots,t_m^{\\pm1}]\\ot\\Lam_n$ and $\\dot\\fk={\\rm Der}(\\dot A)$, there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over $(\\dot A,\\dot\\fk)$ and simple cuspidal Harish-Chndra $A\\fk$-modules for $A:=\\bbbc[t_0^{\\pm1}]\\ot \\dot A$ and $\\fk:={\\rm Der}(A).$ We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair $(\\dot A,\\dot\\fk)$ and show that each such module is a tensor module $\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over (dot A, dot fk) and simple cuspidal Harish-Chandra A fk-modules for A:= C[t0^{pm1}] ot dot A and fk:= Der(A). We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair (dot A, dot fk) and show that each such module is a tensor module dot A ot Omega for an admissible gl(m+1,n)-module Omega via a prescribed action.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The specific algebraic setup with dot A = C[t1^{pm1},...,tm^{pm1}] ot Lambda_n and the restriction to cuspidal simple modules; the correspondence and classification are stated only for this choice of Lie-Rinehart pair and module class, so the result depends on these structural choices holding exactly as defined.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Defines quasi-Poisson modules over Lie-Rinehart pairs and establishes a bijection with Harish-Chandra modules, classifying simple cuspidal examples as tensor modules over gl(m+1,n).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"78e86be20dc58ad0f690e3dd87e14b665a9d67fc1609d8ea2e66b93f79dbbf32"},"source":{"id":"2605.16950","kind":"arxiv","version":1},"verdict":{"id":"10efc6e9-8ed4-4eda-963a-6bd5bdb65a18","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:59:10.818499Z","strongest_claim":"there is a one-to-one correspondence between simple cuspidal quasi-Poisson modules over (dot A, dot fk) and simple cuspidal Harish-Chandra A fk-modules for A:= C[t0^{pm1}] ot dot A and fk:= Der(A). We also classify simple cuspidal quasi-Poisson modules over the Lie-Rinehart pair (dot A, dot fk) and show that each such module is a tensor module dot A ot Omega for an admissible gl(m+1,n)-module Omega via a prescribed action.","one_line_summary":"Defines quasi-Poisson modules over Lie-Rinehart pairs and establishes a bijection with Harish-Chandra modules, classifying simple cuspidal examples as tensor modules over gl(m+1,n).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The specific algebraic setup with dot A = C[t1^{pm1},...,tm^{pm1}] ot Lambda_n and the restriction to cuspidal simple modules; the correspondence and classification are stated only for this choice of Lie-Rinehart pair and module class, so the result depends on these structural choices holding exactly as defined.","pith_extraction_headline":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16950/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T19:52:11.225259Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:31:18.920208Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T19:10:41.565141Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T18:41:56.239049Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.322616Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f3acd02164fd51026740cbff1ad1637431a8d296d170a57a99d26d4fd195be17"},"references":{"count":13,"sample":[{"doi":"","year":2022,"title":"Billig, Towards Kac-Van de Leur conjecture: locality of superconfo rmal algebras , Advances in Mathe- matics 400 (2022), 108295","work_id":"96cfa5c8-4f65-47fe-999d-a41e68a66fe7","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Y. Billig, V. Futorny, K. Iohara K. and I. Kashuba, Classification of simple cuspidal strong Harish-Chandra W (m,n )-modules, arXiv preprint arXiv:2006.05618 (2020)","work_id":"2db031af-540a-4530-964e-e7ddda6dea89","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"C. Chen and V. Mazorchuk, Simple supermodules over Lie superalgebras , Transactions of the American Mathematical Society 374(2)(2021), 899–921","work_id":"825efbaf-cbe4-4685-969f-bacd9ee66c0a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2004,"title":"Eswara Rao, Partial classification of modules for Lie-algebra of diffeom orphisms of d-dimensional torus , Journal of mathematical physics 45(8) (2004), 3322–3333","work_id":"9fc7ade6-37f6-4c1c-aaf8-6bb6338beab2","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1990,"title":"Fernando, Lie algebra modules with finite dimensional weight spaces I , Transactions of the American Mathematical Society","work_id":"97bc2660-20bb-44b0-8879-7826cafcf9a3","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":13,"snapshot_sha256":"cc380cc1317148198170f9ce41dbf5c36edb10080b22ac2159d545e7ded1331a","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"16160c00958b8c828adcc879abb18c8da3b221eae3ed37f83acdbdafd83c5229"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}