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

TL;DR

This paper classifies 2-generated primitive axial algebras of Monster type $(2eta, eta)$, showing they are generated by 8 vectors over rings with invertible 2 and beta, and provides a complete classification over fields of characteristic not 2.

Contribution

It proves that such algebras are generated by 8 vectors and offers a complete classification over fields with characteristic not 2.

Findings

Generated as R-module by 8 vectors

Complete classification over fields of characteristic not 2

Results hold over rings with invertible 2 and 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 $2$.