On Griess Algebras

In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V = V_0 \oplus V_2 \oplus V_3\oplus …$,suchthat$\dim V_0 = 1$and$V_2$contains$A$\mathrm{.}Wecanchoose$V$sothatif$A$hasaunit$e$,then$2e$istheVirasoroelementof$V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.