{"paper":{"title":"On the local structure of noncommutative deformations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Mohamed Boucetta, Zouhair Saassai","submitted_at":"2014-01-02T17:28:48Z","abstract_excerpt":"Let $(M,\\pi,\\mathcal{D})$ be a Poisson manifold endowed with a flat, torsion-free contravariant connection. We show that if $\\mathcal{D}$ is an $\\mathcal{F}$-connection then there exists a tensor $\\mathbf{T}$ such that $\\mathcal{D}\\mathbf{T}$ is the metacurvature tensor introduced by E. Hawkins in his work on noncommutative deformations. We compute $\\mathbf{T}$ and the metacurvature tensor in this case, and show that if $\\mathbf{T}=0$ then, near any regular point, $\\pi$ and $\\mathcal{D}$ are defined in a natural way by a Lie algebra action and a solution of the classical Yang-Baxter equation. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.0477","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":""},"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"}