A New Division Algebra Representation of $E_7$

TL;DR

This paper introduces a novel division algebra-based matrix representation of the exceptional Lie algebra $E_7$, revealing explicit structural decompositions within the larger algebra $E_8$ using division algebra operations.

Contribution

It provides a new explicit matrix-based realization of $E_7$ within $E_8$ using division algebras, enhancing understanding of their algebraic structures.

Findings

Decomposition of $rak{e}_8$ into $rak{e}_7$ and $rak{sl}(2,b R)$ representations.

Explicit realization of Freudenthal's description within $rak{e}_8$.

Implementation of algebraic operations like trace, Freudenthal product, and determinant via commutators.

Abstract

We decompose the Lie algebra $\mathfrak{e}_{8(-24)}$ into representations of $\mathfrak{e}_{7(-25)}\oplus\mathfrak{sl}(2,\mathbb{R})$ using our recent description of $\mathfrak{e}_8$ in terms of (generalized) $3\times3$ matrices over pairs of division algebras. Freudenthal's description of both $\mathfrak{e}_7$ and its minimal representation are therefore realized explicitly within $\mathfrak{e}_8$, with the action given by the (generalized) matrix commutator in $\mathfrak{e}_8$, and with a natural parameterization using division algebras. Along the way, we show how to implement standard operations on the Albert algebra such as trace of the Jordan product, the Freudenthal product, and the determinant, all using commutators in $\mathfrak{e}_8$.