Primitive axial algebras are of Jordan type

TL;DR

This paper proves that all algebras generated by primitive axes, regardless of their type, are classified as primitive axial algebras of Jordan type, linking algebraic structures to group theory and vertex operator algebras.

Contribution

It completes the proof that primitive axial algebras generated by primitive axes are of Jordan type, extending previous results to more general cases.

Findings

All algebras generated by primitive axes are of Jordan type.

The work connects axial algebras to 3-transposition groups and vertex operator algebras.

Provides a comprehensive classification of primitive axial algebras.

Abstract

The notion of axial algebra is closely related to $3$-transposition groups, the Monster group and vertex operator algebras. In this work we continue our previous works and compete the proof that all algebras generated by a set of primitive axes not necessarily of the same type (see the definition in the body of the paper), are primitive axial algebras of Jordan type.