Weakly primitive axial algebras

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 various noncommutative 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 PAJ's), and we see, in contrast to the case for~PAJ's, that there are higher dimensional weak PAJ's generated by two axes. We also prove a theorem that enables us to reduce weak PAJ's to uniform components.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar