Composition algebras and the two faces of $G_{2}$

TL;DR

This paper explores the dual roles of the group G2 in relation to octonions and 3-forms in seven dimensions, revealing their equivalence through geometric and algebraic perspectives, with implications for physics.

Contribution

It demonstrates the equivalence of G2's roles as automorphism group of octonions and as isotropy group of a 3-form, using a regular metric and group diagrams.

Findings

G2 acts as automorphism group of octonions.

G2 is the isotropy group of a generic 3-form in 7D.

The two roles are shown to be equivalent via a regular metric.

Abstract

We consider composition and division algebras over the real numbers: We note two r\^oles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.