Weakly primitive axial algebras

TL;DR

This paper extends the study of axial algebras by weakening primitivity conditions, exploring new examples, and characterizing low-dimensional cases, including noncommutative and weakly primitive scenarios.

Contribution

It introduces the concept of weakly primitive axial algebras, broadening the class of studied algebras and providing classifications for 2-dimensional and certain higher-dimensional cases.

Findings

Introduction of weakly primitive axial algebras with generalized fusion rules

Classification of 2-dimensional weak PAJ's and their properties

Existence of weak PAJ's of dimension greater than 3 generated by two axes

Abstract

In earlier work we studied the structure of primitive axial algebras of Jordan type (PAJ's), not necessarily commutative, in terms of their primitive axes. In this paper we weaken primitivity and permit several pairs of (left and right) eigenvalues satisfying a more general fusion rule, bringing in interesting new examples such as the band semigroup algebras and other commutative and noncommutative examples. Also we broaden our investigation and describe 2-generated algebras in which only one of the generating axes is weakly primitive and satisfies the fusion rules, on condition that its zero-eigenspace is one dimensional. We also characterize when both axes satisfy the fusion rules (weak PAJ's), and describe precisely the 2-dimensional axial algebras. In contrast to the previous situation, there are weak PAJ's of dimension~$> 3$ generated by two axes.