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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Jordan-Chevalley decomposition in finite dimensional Lie algebras
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by Leandro Cagliero and Fernando Szechtman PDF
Proc. Amer. Math. Soc. 139 (2011), 3909-3913 Request permission

Abstract:

Let $\mathfrak {g}$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\mathfrak {g}$ is said to have an abstract Jordan-Chevalley decomposition if there exist unique $s,n\in \mathfrak {g}$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $\pi :\mathfrak {g}\to \mathfrak {gl}(V)$ the Jordan-Chevalley decomposition of $\pi (x)$ in $\mathfrak {gl}(V)$ is $\pi (x)=\pi (s)+\pi (n)$.

In this paper we prove that $x\in \mathfrak {g}$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\mathfrak {g},\mathfrak {g}]$, in which case its semisimple and nilpotent parts are also in $[\mathfrak {g},\mathfrak {g}]$ and are explicitly determined. We derive two immediate consequences: (1) every element of $\mathfrak {g}$ has an abstract Jordan-Chevalley decomposition if and only if $\mathfrak {g}$ is perfect; (2) if $\mathfrak {g}$ is a Lie subalgebra of $\mathfrak {gl}(n,k)$, then $[\mathfrak {g},\mathfrak {g}]$ contains the semisimple and nilpotent parts of all its elements. The last result was first proved by Bourbaki using different methods.

Our proof uses only elementary linear algebra and basic results on the representation theory of Lie algebras, such as the Invariance Lemma and Lie’s Theorem, in addition to the fundamental theorems of Ado and Levi.

References
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Additional Information
  • Leandro Cagliero
  • Affiliation: CIEM-CONICET, FAMAF-Universidad Nacional de Córdoba, Córdoba, Argentina
  • Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
  • Email: cagliero@famaf.unc.edu.ar
  • Fernando Szechtman
  • Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2
  • Email: fernando.szechtman@gmail.com
  • Received by editor(s): July 5, 2010
  • Received by editor(s) in revised form: September 29, 2010, and October 7, 2010
  • Published electronically: March 31, 2011
  • Additional Notes: The first author was supported in part by CONICET and SECYT-UNC grants.
    The second author was supported in part by an NSERC discovery grant
    Both authors thank A. Pianzola for pointing out the work of Bourbaki on decomposable Lie algebras and J. Vargas for useful discussions on the paper.
  • Communicated by: Gail R. Letzter
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 3909-3913
  • MSC (2010): Primary 17B05, 17B10; Secondary 15A21
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10827-X
  • MathSciNet review: 2823037