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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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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)."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules."}],"snapshot_sha256":"78e86be20dc58ad0f690e3dd87e14b665a9d67fc1609d8ea2e66b93f79dbbf32"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"16160c00958b8c828adcc879abb18c8da3b221eae3ed37f83acdbdafd83c5229"},"integrity":{"available":true,"clean":true,"detectors_run":[{"findings_count":0,"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"}],"endpoint":"/pith/2605.16950/integrity.json","findings":[],"snapshot_sha256":"f3acd02164fd51026740cbff1ad1637431a8d296d170a57a99d26d4fd195be17","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 $\\","authors_text":"Malihe Yousofzadeh","cross_cats":[],"headline":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules.","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.RT","submitted_at":"2026-05-16T12:02:34Z","title":"Quasi-Poisson Modules and Harish-Chandra $\\bs{AD}$-Modules"},"references":{"count":13,"internal_anchors":0,"resolved_work":13,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"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","year":2022},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":2006},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"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","year":2021},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"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","year":2004},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Fernando, Lie algebra modules with finite dimensional weight spaces I , Transactions of the American Mathematical Society","work_id":"97bc2660-20bb-44b0-8879-7826cafcf9a3","year":1990}],"snapshot_sha256":"cc380cc1317148198170f9ce41dbf5c36edb10080b22ac2159d545e7ded1331a"},"source":{"id":"2605.16950","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T18:59:10.818499Z","id":"10efc6e9-8ed4-4eda-963a-6bd5bdb65a18","model_set":{"reader":"grok-4.3"},"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","pith_extraction_headline":"Simple cuspidal quasi-Poisson modules over a Lie-Rinehart pair correspond one-to-one with simple cuspidal Harish-Chandra modules.","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.","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."}},"verdict_id":"10efc6e9-8ed4-4eda-963a-6bd5bdb65a18"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:8edc28082873a253139da935c29416d99cd81348d996ffa1bc70eceb45cf02c3","target":"record","created_at":"2026-05-20T00:03:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"69ef24d8e56bd6209a826f483e0e502cec6869fd0fc2be5b9734b9bbfb65a094","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.RT","submitted_at":"2026-05-16T12:02:34Z","title_canon_sha256":"9382b78e7ff959cf7c664540e2147855f83aeef61e8ac02310e5ac90bc313ff1"},"schema_version":"1.0","source":{"id":"2605.16950","kind":"arxiv","version":1}},"canonical_sha256":"004e138ac1d41aa239a61765a5d94398c06400a855246fef7a1f1f735d37e0de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"004e138ac1d41aa239a61765a5d94398c06400a855246fef7a1f1f735d37e0de","first_computed_at":"2026-05-20T00:03:32.484307Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:03:32.484307Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BQIjVqBvI5zH8euMGGMhSNJxYtZ/nj9TzgmeoJBDlzom4PqGRf3LXLOqQFv2Hn0Glty/HUDNtm57FUJP6OSwBA==","signature_status":"signed_v1","signed_at":"2026-05-20T00:03:32.485118Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.16950","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8edc28082873a253139da935c29416d99cd81348d996ffa1bc70eceb45cf02c3","sha256:6f2a669dac3b75255a6d6bd4b11833d6e1bc5427db497610e271591b97c1813a"],"state_sha256":"acec9d2d3030c8c84e0f2da926351c772b3a5fc723335e89b2dacbbf32a2c0f8"}