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

TL;DR

This paper constructs an infinite-dimensional primitive axial algebra of Monster type, demonstrating a notable exception to a previously established dimension bound for certain parameters.

Contribution

It provides the first example of an infinite-dimensional 2-generated primitive axial algebra of Monster type with specific parameters, challenging existing bounds.

Findings

Constructed an infinite-dimensional algebra of Monster type (2, 1/2).

Showed the exception to Rehren's dimension bound for certain parameters.

Demonstrated the existence over arbitrary fields with char ≠ 2,3.

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.