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# Jordan-Chevalley decomposition in Lie algebras

Published:2018-07-23

• Cagliero Leandro,
CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina
• Fernando Szechtman,
Department of Mathematics and Statistics, Univeristy of Regina, Regina, Saskatchewan
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## Abstract

We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.
 Keywords: solvable Lie algebra, Jordan-Chevalley decomposition, representation
 MSC Classifications: 17-08 - Computational methods 17B05 - Structure theory 20C40 - Computational methods 15A21 - Canonical forms, reductions, classification

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