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Octonionic Cayley spinors and $E_{6}$

Tevian Dray; Corinne A. Manogue — 2010

Commentationes Mathematicae Universitatis Carolinae

Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group $E_{6}$ , and of its subgroups. We are therefore led to a description of $E_{6}$ in terms of $3 \times 3$ octonionic matrices, generalizing previous results in the $2 \times 2$ case. Our treatment naturally includes a description of several important subgroups of $E_{6}$ , notably $G_{2}$ , $F_{4}$ , and (the double cover of) $S O (9, 1)$ . An interpretation of the actions...

A symplectic representation of $E_{7}$

Tevian Dray; Corinne A. Manogue; Robert A. Wilson — 2014

Commentationes Mathematicae Universitatis Carolinae

We explicitly construct a particular real form of the Lie algebra $𝔢_{7}$ in terms of symplectic matrices over the octonions, thus justifying the identifications $𝔢_{7} ≅ 𝔰𝔭 (6, 𝕆)$ and, at the group level, $E_{7} ≅ Sp (6, 𝕆)$ . Along the way, we provide a geometric description of the minimal representation of $𝔢_{7}$ in terms of rank 3 objects called cubies.

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Octonionic Cayley spinors and E 6

A symplectic representation of E 7

Octonionic Cayley spinors and $E_{6}$

A symplectic representation of $E_{7}$