{"paper":{"title":"Lie Subalgebras of vector fields and the Jacobian Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Andriy Regeta","submitted_at":"2013-11-01T17:34:07Z","abstract_excerpt":"We study Lie subalgebras $L$ of the vector fields $\\mathrm{Vec}^{c}({\\mathbb A}^{2})$ of affine 2-space ${\\mathbb A}^{2}$ of constant divergence, and we classify those $L$ which are isomorphic to the Lie algebra $\\mathfrak{aff}_{2}$ of the group $\\mathrm{Aff}_{2}(K)$ of affine transformations of ${\\mathbb A}^{2}$. We then show that the following three statements are equivalent: (i) The Jacobian Conjecture holds in dimension 2; (ii) All Lie subalgebras $L \\subset \\mathrm{Vec}^{c}({\\mathbb A}^{2})$ isomorphic to $\\mathfrak{aff}_{2}$ are conjugate under $\\mathrm{Aut}({\\mathbb A}^{2})$; (iii) All "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.0232","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"}