{"paper":{"title":"It\\^o's theorem and metabelian Leibniz algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.RA","authors_text":"A.L. Agore, G. Militaru","submitted_at":"2014-01-19T14:52:18Z","abstract_excerpt":"We prove that the celebrated It\\^{o}'s theorem for groups remains valid at the level of Leibniz algebras: if $\\mathfrak{g}$ is a Leibniz algebra such that $\\mathfrak{g} = A + B$, for two abelian subalgebras $A$ and $B$, then $\\mathfrak{g}$ is metabelian, i.e. $[ \\, [\\mathfrak{g}, \\, \\mathfrak{g}], \\, [ \\mathfrak{g}, \\, \\mathfrak{g} ] \\, ] = 0$. A structure type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimension $1$ are described, classified and their automorphisms groups are explicitly determined as subgroups of a semi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4675","kind":"arxiv","version":3},"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"}