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

DOI:
https://doi.org/10.1090/S0002-9939-2011-10827-X

Published electronically:
March 31, 2011

MathSciNet review:
2823037

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Abstract | References | Similar Articles | Additional Information

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.

- Armand Borel,
*Linear algebraic groups*, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR**1102012** - Nicolas Bourbaki,
*Lie groups and Lie algebras. Chapters 1–3*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998. Translated from the French; Reprint of the 1989 English translation. MR**1728312** - Nicolas Bourbaki,
*Lie groups and Lie algebras. Chapters 7–9*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. MR**2109105** - William Fulton and Joe Harris,
*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249** - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR**499562** - G. B. Seligman,
*Modular Lie algebras*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR**0245627**

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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

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.