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Jordan–Chevalley Decomposition in Lie Algebras

Part of: Basic linear algebra Representation theory of groups Nonassociative rings and algebras Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  28 February 2019

Leandro Cagliero  and
Fernando Szechtman
Leandro Cagliero
Affiliation:
CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina Email: cagliero@famaf.unc.edu.ar
Fernando Szechtman
Affiliation:
Department of Mathematics and Statistics, Univeristy of Regina, Regina, SK Email: fernando.szechtman@gmail.com
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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 $A=S+N$.

Keywords

solvable Lie algebraJordan–Chevalley decompositionrepresentation

MSC classification

Primary: 17-08: Computational methods
Secondary: 17B05: Structure theory 20C40: Computational methods 15A21: Canonical forms, reductions, classification
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 349 - 354
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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