On the structure of axial algebras

TL;DR

This paper develops the structure theory of axial algebras, focusing on radical, simplicity, and sum decompositions, which are key to understanding their algebraic properties and applications.

Contribution

It advances the understanding of axial algebras by analyzing their radical, simplicity, and sum decompositions, providing foundational insights into their structure.

Findings

Characterization of radical and simple axial algebras

Decomposition theorems for axial algebras

Framework for analyzing sum decompositions

Abstract

Axial algebras are a recently introduced class of non-associative algebra motivated by applications to groups and vertex-operator algebras. We develop the structure theory of axial algebras focussing on two major topics: (1) radical and simplicity; and (2) sum decompositions.