An expansion algorithm for constructing axial algebras

TL;DR

This paper introduces a new expansion algorithm for constructing axial algebras with specified Miyamoto groups, overcoming previous limitations and enabling the computation of examples related to the Monster fusion law.

Contribution

The paper presents a novel expansion algorithm for axial algebras that overcomes the 2-closeness restriction of previous methods, facilitating the construction of algebras with given Miyamoto groups.

Findings

Algorithm successfully constructs axial algebras with specified Miyamoto groups.

Implemented in MAGMA, the algorithm computes examples for the Monster fusion law.

Provides a list of new examples demonstrating the algorithm's effectiveness.

Abstract

An axial algebra $A$ is a commutative non-associative algebra generated by primitive idempotents, called axes, whose adjoint action on $A$ is semisimple and multiplication of eigenvectors is controlled by a certain fusion law. Different fusion laws define different classes of axial algebras. Axial algebras are inherently related to groups. Namely, when the fusion law is graded by an abelian group $T$, every axis $a$ leads to a subgroup of automorphisms $T_a$ of $A$. The group generated by all $T_a$ is called the Miyamoto group of the algebra. We describe a new algorithm for constructing axial algebras with a given Miyamoto group. A key feature of the algorithm is the expansion step, which allows us to overcome the $2$-closeness restriction of Seress's algorithm computing Majorana algebras. At the end we provide a list of examples for the Monster fusion law, computed using a MAGMA implementation of our algorithm.