Clifford Elements in Lie Algebras

José Ramón Brox¹ ; Antonio Fernández López¹ ; Miguel Gómez Lozano¹

¹Dep. de Álgebra, Geometría y Topología, Universidad de Málaga, 29071 Málaga, Spain

Journal of Lie Theory, Tome 27 (2017) no. 1, pp. 283-296

Résumé

\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 $\sk(R,*)$.

Détail

Classification : 17B60, 17C50, 16N60
Mots-clés : Lie algebra, ring with involution, Jordan algebra, inner ideal, Jordan element