Journal of Lie Theory 27 (2017), No. 1, 283--296
Copyright Heldermann Verlag 2017
Clifford Elements in Lie Algebras
José Ramón Brox
Dep. de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
brox@agt.cie.uma.es
Antonio Fernández López
Dep. de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
emalfer@uma.es
Miguel Gómez Lozano
Dep. de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain
miggl@uma.es
[Abstract-pdf]
\def\F{\mathbb{F}} \def\sk{{\rm Skew}} \def\ad{\mathop{\rm ad}\nolimits} Let $L$ be a Lie algebra over a field $\F$ of characteristic zero or $p>3$. An element $c\in L$ is called {\it Clifford} if $\ad_c^3=0$ and its associated Jordan algebra $L_c$ is the Jordan algebra $\F \oplus X$ defined by a symmetric bilinear form on a vector space $X$ over $\F$. In this paper we prove the following result: Let $R$ be a centrally closed prime ring $R$ of characteristic zero or $p > 3$ with involution $*$ and let $c\in \sk(R,*)$ be such that $c^3=0$, $c^2 \neq 0$ and $c^2kc =ckc^2$ for all $k \in \sk(R,*)$. Then $c$ is a Clifford element of the Lie algebra $\sk(R,*)$.
Keywords: Lie algebra, ring with involution, Jordan algebra, inner ideal, Jordan element.
MSC: 17B60, 17C50, 16N60
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