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 $(α,β)$ over a field $\mathbb F$. Using this, we first show that every such algebra has dimension at most 8, except for the case $(α,β)=(2,\tfrac{1}{2})$, 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 $(α,β)$ over ${\mathbb Q}(α,β)$, for $α$ and $β$ 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.

Theorems & Definitions (108)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Remark 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof