A New Division Algebra Representation of $E_6$

TL;DR

This paper presents an explicit matrix-based division algebra approach to decompose the Lie algebra e_8 into e_6 and su(3) representations, revealing the Albert algebra within e_8.

Contribution

It introduces a novel explicit matrix representation over division algebras to realize the Albert algebra inside e_8, extending from Lie algebra to Lie group.

Findings

Explicit matrix representation of e_8 decomposition.

Realization of the Albert algebra within e_8.

Natural extension to e_6 Lie group.

Abstract

We construct the well-known decomposition of the Lie algebra $\mathfrak{e}_8$ into representations of $\mathfrak{e}_6\oplus\mathfrak{su}(3)$ using explicit matrix representations over pairs of division algebras. The minimal representation of $\mathfrak{e}_6$, namely the Albert algebra, is thus realized explicitly within $\mathfrak{e}_8$, with the action given by the matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Each resulting copy of the Albert algebra consists of anti-Hermitian matrices in $\mathfrak{e}_8$, labeled by imaginary (split) octonions. Our formalism naturally extends from the Lie algebra to the Lie group $E_6\subset E_8$.