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

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.

12 pages; 2 double figures

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