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

Author:

Mauricio Godoy Molina, Diego Lagos

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Rings and Algebras (math.RA), Representation Theory (math.RT)

journal:

date:

2023-06-04 16:00:00

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 $\mathfrak{g}_2$.

PDF: $|3|-$gradings of complex simple Lie algebras.pdf