On axial algebras with $3$ eigenvalues

TL;DR

This paper investigates axial algebras with three eigenvalues, focusing on their structure and properties for algebras generated by two or three idempotents, expanding understanding of their algebraic behavior.

Contribution

It provides a detailed description and analysis of 2- and 3-generated axial algebras with three eigenvalues under minimal fusion law restrictions.

Findings

Characterization of 2-generated axial algebras

Characterization of 3-generated axial algebras

Identification of key algebraic properties

Abstract

We study axial algebras, that is, commutative non-associative algebras generated by idempotents whose adjoint actions are semisimple and obey a fusion law. Considering the case, when said adjoint actions having $3$ eigenvalues and the fusion law is the least restrictive, we describe $2$-generated and $3$-generated algebras and prove some of their general properties.