{"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"}