An infinite-dimensional 2-generated primitive axial algebra of Monster type

Abstract

Rehren proved that a primitive 2-generated axial algebra of Monster type $(\alpha,\beta)$ has dimension at most eight if $\alpha\notin\{2\beta,4\beta\}$. In this note we construct an infinite-dimensional 2-generated primitive axial algebra of Monster type $(2,\frac{1}{2})$ over an arbitrary field $F$ with $char(F)\neq 2,3$. This shows that the second special case, $\alpha=4\beta$, is a true exception to Rehren's bound.