Graded contractions of the $\mathbb{Z}_2^3$-gradings on the exceptional Lie algebras coming from octonions

TL;DR

This paper classifies 860 nonisomorphic 2^3-graded Lie algebras derived from octonion-based gradings on exceptional Lie algebras, revealing their structural diversity and the combinatorial framework used for classification.

Contribution

It provides a comprehensive classification of graded contractions of 2^3-gradings on exceptional Lie algebras, introducing the concept of generalised nice sets as a key tool.

Findings

All graded contractions are necessarily generic.

Supports correspond to generalised nice sets.

Algebras distinguished by Levi decompositions and centers.

Abstract

A total of 860 nonisomorphic \( \mathbb{Z}_2^3 \)-graded Lie algebras of dimensions 52, 78, 133, and 248 are obtained as graded contractions of the \( \mathbb{Z}_2^3 \)-gradings on the exceptional Lie algebras (excluding \( \mathfrak{g}_2 \)) arising from the octonions. It is shown that all graded contractions of these gradings are necessarily generic. Their supports correspond to a combinatorial object known as a \emph{generalised nice set}, which serves as the main tool in the classification. The resulting algebras are distinguished by their Levi decompositions, the derived series of their radicals, their centers, and other structural features.