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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Jordan-Chevalley decomposition in finite dimensional Lie algebras


Authors: Leandro Cagliero and Fernando Szechtman
Journal: Proc. Amer. Math. Soc. 139 (2011), 3909-3913
MSC (2010): Primary 17B05, 17B10; Secondary 15A21
Published electronically: March 31, 2011
MathSciNet review: 2823037
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10827-X
PII: S 0002-9939(2011)10827-X
Keywords: Perfect and semisimple Lie algebras, Jordan-Chevalley decomposition, representations
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.