2-generated axial algebras of Monster type

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 $(\alpha ,\beta )=(2,\frac{1}{2})$, 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.

40 pages, the paper has been completely revised in order to make it easier to read. We improved the results of the previous versions, proving the existence of a bound for the dimension on the algebra also in the case $\alpha =4\beta$, $\alpha \ne 2$