{"paper":{"title":"Jordan-Chevalley decomposition in finite dimesional Lie algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Fernando Szechtman, Leandro Cagliero","submitted_at":"2010-08-06T15:07:49Z","abstract_excerpt":"Let $\\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\\g$ is said to have an \\emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\\in\\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $\\pi:\\g\\to\\gl(V)$ the Jordan-Chevalley decomposition of $\\pi(x)$ in $\\gl(V)$ is $\\pi(x)=\\pi(s)+\\pi(n)$.\n  In this paper we prove that $x\\in\\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\\in [\\g,\\g]$, in which case its semisimple and nilpotent parts are also in $[\\g,\\g]$ and are explicitly dete"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.1217","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"}