Journal of Lie Theory

no. 1
Clifford Elements in Lie Algebras
Journal of Lie Theory, Volume 27 (2017) no. 1, pp. 283-296
\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 $\sk(R,*)$.
DOI: 10.5802/jolt.946
Classification: 17B60, 17C50, 16N60
Keywords: Lie algebra, ring with involution, Jordan algebra, inner ideal, Jordan element 
@article{JOLT_2017_27_1_a15,
     author = {J. R. Brox and A. Fern\'andez L\'opez and M. G\'omez Lozano},
     title = {Clifford {Elements} in {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {283--296},
     year = {2017},
     volume = {27},
     number = {1},
     doi = {10.5802/jolt.946},
     zbl = {1386.17024},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.946/}
}
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J. R. Brox; A. Fernández López; M. Gómez Lozano. Clifford Elements in Lie Algebras. Journal of Lie Theory, Volume 27 (2017) no. 1, pp. 283-296. doi: 10.5802/jolt.946

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