2-generated axial algebras of Monster type

TL;DR

This paper classifies all 2-generated axial algebras of Monster type over various fields, establishing dimension bounds, explicit classifications, and a generalization of the Norton-Sakuma Theorem.

Contribution

It provides the first comprehensive classification of 2-generated axial algebras of Monster type, including dimension bounds, explicit classifications over certain fields, and a generalized Norton-Sakuma Theorem.

Findings

Every such algebra has dimension at most 8, except for a special case with infinite-dimensional examples.

Complete classification over ${ m Q}( extstylerac{1}{4},rac{1}{32})$ for algebraically independent parameters.

Generalization of the Norton-Sakuma Theorem without the Frobenius form hypothesis.

Abstract

We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type $(\alpha,\beta)$ over a field $\mathbb F$. Using this, we first show that every such algebra has dimension at most 8, except for the case $(\alpha,\beta)=(2,\tfrac{1}{2})$, where the Highwater algebra provides examples of dimension $n$, for all $n\in {\mathbb N}\cup \{\infty\}$. We then classify all 2-generated axial algebras of Monster type $(\alpha,\beta)$ over ${\mathbb Q}(\alpha,\beta)$, for $\alpha$ and $\beta$ algebraically independent over $\mathbb Q$. Finally, we generalise the Norton-Sakuma Theorem to every primitive $2$-generated axial algebra of Monster type $(\frac{1}{4},\frac{1}{32})$ over a field of characteristic zero, dropping the hypothesis on the existence of a Frobenius form.