Constructing Chayet-Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for $E_8$)

Abstract

In 2021, Maurice Chayet and Skip Garibaldi provided an explicit construction of a commutative non-associative algebra on the second smallest representation of $E_8$ (of dimension $3875$) adjoined with a unit. In fact, they define such an algebra $A(\mathfrak{g})$ for each simple Lie algebra $\mathfrak{g}$, in terms of explicit but ad-hoc formulas. We discovered that their algebras $A(\mathfrak{g})$ have a natural interpretation in terms of affine vertex algebras, and their ad-hoc formulas take an extremely simple form in this new interpretation. It is our hope that this point of view will lead to a better understanding of this interesting class of algebras.