$|3|-$gradings of complex simple Lie algebras

Abstract

The aim of this paper is to investigate the algebraic structure that appears on $|3|-$gradings $\mathfrak{n}={\mathfrak{n}}_{-3}\oplus \cdots \oplus {\mathfrak{n}}_3$ of a complex simple Lie algebra $\mathfrak{n}$. In particular, we completely determine the possible reductive algebras ${\mathfrak{n}}_0$ and prove that the only free nilpotent Lie algebra of step 3 that appears as the negative part ${\mathfrak{n}}_{-3}\oplus {\mathfrak{n}}_{-2}\oplus {\mathfrak{n}}_{-1}$ of a grading is the usual $|3|-$grading of the exceptional Lie algebra ${\mathfrak{g}}_2$.

15 pages, 6 tables