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 (a₀, a₁) and (b₀, b₁) such that each pair generates a 3A-algebra U_3A and their intersection contains the W₃-algebra \mathcal{W}(4/5) ≅ L(4/5,0) ⊕ 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.