Non-associative Frobenius algebras of type $G_2$ and $F_4$

TL;DR

This paper explicitly describes certain non-associative, commutative algebras associated with groups of type G2 and F4, linking them to octonion and Albert algebras, and determines their automorphism groups.

Contribution

It provides explicit descriptions of these algebras for G2 and F4 types and establishes their automorphism groups as the groups themselves.

Findings

Explicit descriptions using octonion and Albert algebras

Classification of invariant algebra products

Automorphism groups equal to the original groups

Abstract

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2\omega_1$ for $G_2$ and on the representation with highest weight $2\omega_4$ for $F_4$. It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$.