Decompositions of Griess algebras beyond the OZ-setting

Abstract

To each VOA of strong CFT type, we associate a non-associative algebra which generalizes Griess algebras of One-Zero (OZ) VOAs. We compute the fusion law of Ising vectors in this algebra using techniques from Miyamoto, leveraging the module theory of the corresponding Virasoro VOA. This new class of algebras encapsulates the Chayet-Garibaldi algebras, a family of algebras constructed from absolutely simple linear algebraic groups that includes the $3876$-dimensional algebra for $E_8$.