2-generated axial algebras of Monster type $(2\beta ,\beta )$

Abstract

In this paper we prove that $2$-generated primitive axial algebras of Monster type $(2\beta ,\beta )$ over a ring $R$ in which $2$ and $\beta$ are invertible can be generated as $R$-module by $8$ vectors. We then completely classify $2$-generated primitive axial algebras of Monster type $(2\beta ,\beta )$ over any field of characteristic other than $2$.

31 pages. We added the discussion whether a symmetric 2-generated primitive axial algebra of Monster type $(2\beta ,\beta )$ can also be non-symmetric with respect to a different pair of generating axes. This is possible only in the case of the algebra $3C(2/3)$. We also corrected a mistake in Theorem 5.7