A note on 2-generated symmetric axial algebras of Monster type

TL;DR

This paper constructs a new infinite-dimensional symmetric axial algebra of Monster type over a field of characteristic 5, completing the classification of 2-generated symmetric axial algebras of Monster type.

Contribution

It introduces a novel infinite-dimensional algebra in a specific Monster type case, resolving an open classification problem.

Findings

Constructed a new infinite-dimensional algebra of Monster type (2, 1/2)

Completed the classification of primitive symmetric 2-generated axial algebras of Monster type

Demonstrated existence over a field of characteristic 5

Abstract

Recently Takahiro Yabe gave an almost complete classification of primitive symmetric $2$-generated axial algebras of Monster type. In this note, we construct a new infinite-dimensional primitive $2$-generated symmetric axial algebra of Monster type $(2, \frac{1}{2})$ over a field of characteristic $5$, and use this algebra to complete the last case left open.