Constructing Chayet-Garibaldi algebras from affine vertex algebras (including the 3876-dimensional algebra for $E_8$)
Tom De Medts, Louis Olyslager
TL;DR
This work recasts the Chayet–Garibaldi algebras $A(\mathfrak{g}, \kappa)$ as substructures of the simple affine vertex algebra $L_{\widehat{\mathfrak{g}}}(1,0)$, providing a conceptual and uniform framework that clarifies the original ad hoc formulas. By identifying a degree-2 symmetric subspace $L_{(2)}^{\mathrm{sym}}$ endowed with the Jordan product $d \bullet e = \tfrac12(d_1 e + e_1 d)$ and the bilinear form $(d,e) \mapsto d_3 e$, the authors establish an isomorphism $L_{(2)}^{\mathrm{sym}} \cong A(\mathfrak{g}, \kappa)$ (up to scaling of the form) and show $\ker S$ aligns with the maximal graded ideal, enabling the construction inside the quotient $L = V_{\widehat{\mathfrak{g}}}(1,0)/I_{\widehat{\mathfrak{g}}}(1,0)$. The approach yields a natural Frobenius form, a unital criterion, and a central charge formula, and it offers a structural lens for the notable $E_8$ case of dimension $3875$ plus a unit. Overall, the paper bridges invariant-theoretic algebras from algebraic groups with affine vertex algebras, tying non-associative Frobenius algebras to affine Lie theory via a VOA perspective.
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.