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

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.