Primitive axial algebras of Jordan type

TL;DR

This paper classifies primitive axial algebras of Jordan type generated by two elements, focusing on cases where the minimal polynomial divides (x-1)x(x-η), with applications to Griess and Majorana algebras.

Contribution

It provides a classification of 2-generated primitive axial algebras of Jordan type for specific parameters, extending understanding of their automorphisms and structure.

Findings

Classification of 2-generated primitive axial algebras of Jordan type.

Identification of Miyamoto involutions as 3-transpositions for η ≠ 1/2.

Connection to Griess algebras and Majorana algebras.

Abstract

An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.