Simplicity of a vertex operator algebra whose Griess algebra is the Jordan algebra of symmetric matrices

Abstract

Let $r \in \BC$ be a complex number, and $d \in \BZ_{\ge 2}$ a positive integer greater than or equal to 2. Ashihara and Miyamoto introduced a vertex operator algebra $\Vam$ of central charge $dr$, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size $d$. In this paper, we prove that the vertex operator algebra $\Vam$ is simple if and only if $r$ is not an integer. Further, in the case that $r$ is an integer (i.e., $\Vam$ is not simple), we give a generator system of the maximal proper ideal $I_{r}$ of the VOA $\Vam$ explicitly.