From forbidden configurations to a classification of some axial algebras of Monster type

TL;DR

This paper introduces the concepts of axets and shapes to classify certain axial algebras of Monster type, revealing that most such structures collapse, thus extending the understanding of their configurations.

Contribution

It generalizes the shape concept from Majorana algebras to axets, providing a classification of algebras of Monster type and analyzing their collapse behavior.

Findings

Most shapes of generalised Monster type collapse

Classification of all completion algebras for these shapes

Extension of shape theory to axets

Abstract

Ivanov introduced the shape of a Majorana algebra as a record of the $2$-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper.