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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","authors_text":"Cunguang Cheng, Kaiming Zhao, Rencai Lu, Shiyuan Liu, Yueqiang Zhao, Zhiqiang Li","cross_cats":["math.QA","math.RA"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-23T03:25:22Z","title":"Simple smooth modules over the Lie algebras of polynomial vector fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.18262","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:904f6d7fa6a5d78658def4c6bc7e7a704cbb4fe9dca30b6911753f82967f1312","target":"record","created_at":"2026-07-05T11:25:47Z","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":"3d1376c14b746be3c00f0e35b0ad96e1fd27a1f5b044f570692691b7d7f1c49a","cross_cats_sorted":["math.QA","math.RA"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2025-06-23T03:25:22Z","title_canon_sha256":"c7ae175cf7f45738bee228c01975dc0f57b45b927894db41de7ce0110c42b262"},"schema_version":"1.0","source":{"id":"2506.18262","kind":"arxiv","version":1}},"canonical_sha256":"73b7e87c6e37fdbdaec623790d5da5b10d5081f3d48e1f7edc18276858b77bea","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"73b7e87c6e37fdbdaec623790d5da5b10d5081f3d48e1f7edc18276858b77bea","first_computed_at":"2026-07-05T11:25:47.225680Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:25:47.225680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5y0Es6FKamYqMvP4g0MCihz5WUb3eYwZvgb6VDYu+3D+Er9R6FGARgCRSzyO14HL8hSjBSM+9CIa1h3r057aDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T11:25:47.226220Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.18262","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:904f6d7fa6a5d78658def4c6bc7e7a704cbb4fe9dca30b6911753f82967f1312","sha256:c63e37038064c18494769e36bb92927ec64e51c671837c26b78a09d73c8dfb06"],"state_sha256":"2b3a2919b18dcf616c89eb2375416bf4a0de3e69e562672f2f425128156532b7"}