Radicals in primitive axial algebras

TL;DR

This work analyzes the interplay between three radicals in primitive axial algebras with a Frobenius form: the axial radical $R(A)$, the Frobenius form radical $A^{\perp}$, and the Jacobson radical $J(A)$. It proves the inclusions $R(A)\subseteq J(A)\subseteq A^{\perp}$ and shows that if all generating axes are non-singular, then $R(A)=J(A)=A^{\perp}$; it further develops a block/decomposition framework and proves semisimplicity of the form factor and, under non-degenerate Frobenius forms, a direct-sum decomposition into simple primitive axial algebras. The hull-kernel topology on maximal ideals is shown to be discrete, and several open questions are posed regarding the universality of radical equalities and the structure of blocks and domination. These results connect the algebraic structure of axial algebras with their representation-theoretic properties and pave the way for further understanding of their radical theory and decomposition behavior.

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 $A$.

Figures (2)

• Figure 1: Jordan type fusion law $\mathcal{J}(\eta)$
• Figure 2: Monster type fusion law $\mathcal{M}(\alpha,\beta)$

Theorems & Definitions (47)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• Definition 2.7
• Lemma 2.8
• Lemma 2.9