In earlier work, we studied the structure of primitive axial algebras of Jordan type (PAJs), 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 various non-commutative examples. Also, we broaden our investigation to the case of 2-generated algebras for which only one axis satisfies the fusion rules. As an example we describe precisely the 2-dimensional axial algebras and the 3-dimensional and 4-dimensional weakly primitive axial algebras of Jordan type (weak PAJs), and we see, in contrast to the case for PAJs, that there are higher-dimensional weak PAJs generated by two axes. We also prove a theorem that enables us to reduce weak PAJs to uniform components.
This video is part of the European Non-Associative Algebra Seminar series.
