Griess algebras generated by the Griess algebras of two $3A$-algebras with a common axis

Abstract

In this article, we study Griess algebras generated by two pairs of Ising vectors $(a_0, a_1)$ and $(b_0,b_1)$ such that each pair generates a $3A$-algebra $U_{3A}$ and their intersection contains the $W_3$-algebra $\mathcal{W}(4/5)\cong L(4/5,0)\oplus L(4/5,3)$. We show that there are only 3 possibilities, up to isomorphisms and they are isomorphic to the Griess algebras of the VOAs $V_{F(1A)}$, $V_{F(2A)}$ and $V_{F(3A)}$ constructed by Höhn–Lam–Yamauchi.