The Classification of the 2-generated Primitive Axial Algebras of Monster Type

TL;DR

This paper completes the classification of 2-generated primitive axial algebras of Monster type, providing explicit bases, multiplication rules, and consolidating existing results.

Contribution

It finalizes the classification of these algebras, covering all remaining cases and providing detailed algebraic structures.

Findings

Complete classification of 2-generated primitive axial algebras of Monster type.

Explicit bases and multiplication rules for the classified algebras.

Consolidation of existing results into a comprehensive framework.

Abstract

Axial algebras of Monster type are a class of commutative algebras generated by special idempotents called axes. Some motivating examples of these algebras are the Griess algebra and the Norton-Sakuma algebras, relating to the Monster simple group. A long standing open problem is to classify the 2-generated axial algebras of Monster type. A huge milestone was accomplished by Yabe leading, with additional cases completed by Franchi, Mainardis, and McInroy, to the classification in the symmetric case. In this paper, we complete the classification. To do so, we split the proof into multiple cases: dealing with certain parameters, subalgebras, axets, and axial dimensions. Furthermore, we provide a basis, multiplication and information of the algebras in the classification; consolidating existing results on these algebras into one place.