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Clearly, $\\mathfrak{g}$ is a proper subalegbra of $\\mathcal{L}$.\n  Surprisingly, we prove that simple smooth modules over $\\mathfrak{g}$ are exactly the simple modules over $\\mathcal{L}$ studied by Rodakov (no need to take completion). Then we find an easy and elementary way to classify all simple smooth modules over $\\mathfrak{g}$. When the height $\\ell_{V}\\geq2$ or $n=1$, any nontrivial simple smooth\n  $\\mathfrak{g}$-module $V$ is isomorph"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2506.18262","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-23T03:25:22Z","cross_cats_sorted":["math.QA","math.RA"],"title_canon_sha256":"c7ae175cf7f45738bee228c01975dc0f57b45b927894db41de7ce0110c42b262","abstract_canon_sha256":"3d1376c14b746be3c00f0e35b0ad96e1fd27a1f5b044f570692691b7d7f1c49a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:47.226220Z","signature_b64":"5y0Es6FKamYqMvP4g0MCihz5WUb3eYwZvgb6VDYu+3D+Er9R6FGARgCRSzyO14HL8hSjBSM+9CIa1h3r057aDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"73b7e87c6e37fdbdaec623790d5da5b10d5081f3d48e1f7edc18276858b77bea","last_reissued_at":"2026-07-05T11:25:47.225680Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:47.225680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Simple smooth modules over the Lie algebras of polynomial vector fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.QA","math.RA"],"primary_cat":"math.RT","authors_text":"Cunguang Cheng, Kaiming Zhao, Rencai Lu, Shiyuan Liu, Yueqiang Zhao, Zhiqiang Li","submitted_at":"2025-06-23T03:25:22Z","abstract_excerpt":"Let $\\mathfrak{g}:={\\rm Der}(\\mathbb{C}[t_1, t_2,\\cdots, t_n])$ and $\\mathcal{L}:={\\rm Der}(\\mathbb{C}[[t_1, t_2,\\cdots, t_n]])$ be the Witt Lie algebras. Clearly, $\\mathfrak{g}$ is a proper subalegbra of $\\mathcal{L}$.\n  Surprisingly, we prove that simple smooth modules over $\\mathfrak{g}$ are exactly the simple modules over $\\mathcal{L}$ studied by Rodakov (no need to take completion). Then we find an easy and elementary way to classify all simple smooth modules over $\\mathfrak{g}$. When the height $\\ell_{V}\\geq2$ or $n=1$, any nontrivial simple smooth\n  $\\mathfrak{g}$-module $V$ is isomorph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.18262","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.18262/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":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.18262","created_at":"2026-07-05T11:25:47.225739+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.18262v1","created_at":"2026-07-05T11:25:47.225739+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.18262","created_at":"2026-07-05T11:25:47.225739+00:00"},{"alias_kind":"pith_short_12","alias_value":"OO36Q7DOG763","created_at":"2026-07-05T11:25:47.225739+00:00"},{"alias_kind":"pith_short_16","alias_value":"OO36Q7DOG7633LWG","created_at":"2026-07-05T11:25:47.225739+00:00"},{"alias_kind":"pith_short_8","alias_value":"OO36Q7DO","created_at":"2026-07-05T11:25:47.225739+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE","json":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE.json","graph_json":"https://pith.science/api/pith-number/OO36Q7DOG7633LWGEN4Q2XNFWE/graph.json","events_json":"https://pith.science/api/pith-number/OO36Q7DOG7633LWGEN4Q2XNFWE/events.json","paper":"https://pith.science/paper/OO36Q7DO"},"agent_actions":{"view_html":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE","download_json":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE.json","view_paper":"https://pith.science/paper/OO36Q7DO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.18262&json=true","fetch_graph":"https://pith.science/api/pith-number/OO36Q7DOG7633LWGEN4Q2XNFWE/graph.json","fetch_events":"https://pith.science/api/pith-number/OO36Q7DOG7633LWGEN4Q2XNFWE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE/action/storage_attestation","attest_author":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE/action/author_attestation","sign_citation":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE/action/citation_signature","submit_replication":"https://pith.science/pith/OO36Q7DOG7633LWGEN4Q2XNFWE/action/replication_record"}},"created_at":"2026-07-05T11:25:47.225739+00:00","updated_at":"2026-07-05T11:25:47.225739+00:00"}