Radicals in primitive axial algebras

TL;DR

This paper investigates the structure of primitive axial algebras with Frobenius forms, focusing on comparing key radicals and ideals such as the largest ideal not containing axes, the radical of the form, and the Jacobson radical.

Contribution

It introduces a comparison of different radicals and ideals in primitive axial algebras with Frobenius forms, enhancing the understanding of their internal structure.

Findings

Comparison of the largest ideal $R(A)$ and the radical $A^ot$

Analysis of the Jacobson radical $J(A)$ in primitive axial algebras

Insights into the structure theory of primitive axial algebras

Abstract

The paper contributes to the structure theory of primitive axial algebras. For a primitive axial algebra $A$ with a Frobenius form we compare the largest ideal $R(A)$ not containing any of the generating axes, the radical $A^\perp$ of the form, and the Jacobson radical $J(A)$, which we define simply as the intersection of all maximal ideals of $A$.