{"paper":{"title":"Poisson enveloping algebras and the Poincar\\'e-Birkhoff-Witt theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Cyrille Ospel, Pol Vanhaecke, Thierry Lambre","submitted_at":"2016-01-07T13:20:19Z","abstract_excerpt":"Poisson algebras are, just like Lie algebras, particular cases of Lie-Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves --- under some mild conditions --- that the enveloping algebra of a Lie-Rinehart algebra satisfies a Poincar\\'e-Birkhoff-Witt theorem (PBW theorem). In the case of a Poisson algebra $({\\mathcal A},\\cdot,\\{\\cdot,\\cdot\\})$ over a commutative ring $R$ (with unit), Rinehart's result boils down to the statement that if $\\mathcal A$ is \\emph{smooth} (as an algebra), then gr$(U"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.01528","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}