Enumerating 3-generated axial algebras of Monster type

TL;DR

This paper classifies a specific subclass of 3-generated axial algebras of Monster type, revealing most potential configurations collapse, and introduces a method to efficiently identify such collapsing shapes.

Contribution

It provides the first enumeration of 3-generated axial algebras of Monster type and develops a minimal forbidden configurations method for recognizing collapsing shapes.

Findings

Most potential shapes for 3-generated axial algebras of Monster type collapse.

The enumeration reveals a limited set of non-trivial examples.

A new method for recognizing collapsing shapes is introduced.

Abstract

An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The $2$-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of $3$-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.