A Symplectic Representation of $\mathrm{E}_7$

TL;DR

This paper constructs a symplectic matrix representation of the exceptional Lie algebra E7 over octonions, providing geometric insights into its minimal representation and confirming algebraic and group isomorphisms.

Contribution

It explicitly constructs a real form of 7 as symplectic matrices over octonions, clarifying algebraic and group isomorphisms and offering a geometric perspective on its minimal representation.

Findings

Explicit symplectic matrix construction of 7 over octonions

Identification of 7 with 6(\u2205) in symplectic form

Geometric description of the minimal representation using cubies

Abstract

We explicitly construct a particular real form of the Lie algebra $\mathfrak{e}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak{e}_7\cong\mathfrak{sp}(6,\mathbb{O})$ and, at the group level, $\mathrm{E}_7\cong\mathrm{Sp}(6,\mathbb{O})$. Along the way, we provide a geometric description of the minimal representation of $\mathfrak{e}_7$ in terms of rank 3 objects called cubies.