A guide to the $2$-generated axial algebras of Monster type

TL;DR

This paper thoroughly analyzes the properties of 2-generated axial algebras of Monster type, including their ideals, subalgebras, and isomorphisms, providing new bases and computational tools to understand their structure.

Contribution

It offers a detailed classification and analysis of all 2-generated axial algebras of Monster type, including new bases and computational resources.

Findings

Classification of 12 infinite families and exceptional algebras

Descriptions of ideals, quotients, and subalgebras

Identification of all exceptional isomorphisms

Abstract

Axial algebras of Monster type are a class of non-associative algebras which generalise the Griess algebra, whose automorphism group is the largest sporadic simple group, the Monster. The $2$-generated algebras, which are the building blocks from which all algebras in this class can be constructed, have recently been classified by Yabe; Franchi and Mainardis; and Franchi, Mainardis and McInroy. There are twelve infinite families of examples as well as the exceptional Highwater algebra and its cover, however their properties are not well understood. In this paper, we detail the properties of each of these families, describing their ideals and quotients, subalgebras and idempotents in all characteristics. We also describe all exceptional isomorphisms between them. We give new bases for several of the algebras which better exhibit their axial features and provide code for others to work with them.