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Jordan–Chevalley Decomposition in Lie Algebras

Published online by Cambridge University Press:  28 February 2019

Leandro Cagliero
Affiliation:
CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina Email: [email protected]
Fernando Szechtman
Affiliation:
Department of Mathematics and Statistics, Univeristy of Regina, Regina, SK Email: [email protected]
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Abstract

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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 $A=S+N$.

Type
Article
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author L. C. was supported in part by CONICET and SECYT-UNC grants.

Author F. S. was supported in part by an NSERC discovery grant.

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