Linear models of the exceptional Lie algebra $\mathfrak{e}_8$

TL;DR

This paper introduces five explicit, graded models of the exceptional Lie algebra e_8, constructed from its maximal rank semisimple subalgebras, enhancing understanding of its algebraic structure.

Contribution

It provides novel explicit constructions of e_8 based on graded decompositions, which are new representations of this complex algebra.

Findings

Five explicit models of e_8 are constructed.

Each model is graded by a different abelian group.

The neutral component is a sum of special linear algebras.

Abstract

This work provides five explicit constructions of the exceptional Lie algebra $\mathfrak{e}_8$, based on its semisimple subalgebras of maximal rank. Each of these models is graded by an abelian group, namely, $\mathbb{Z}_4$, $\mathbb{Z}_5$, $\mathbb{Z}_6$, $\mathbb{Z}_3^2$ and $\mathbb{Z}_2\times\mathbb{Z}_4$; the neutral component is direct sum of special linear algebras and the remaining homogeneous components are irreducible modules for it.