Linear models of the exceptional Lie algebra e8

Abstract

This work presents five explicit constructions of the exceptional Lie algebra e8, each associated with a semisimple subalgebra of maximal rank. The provided models are based on gradings by finite abelian groups, namely, Z4, Z5, Z6, Z2 3, and Z2 ×Z4. In all cases, the neutral component is a direct sum of special linear algebras, while the remaining homogeneous components are irreducible modules over it.