Axes in non-associative algebras

TL;DR

This paper explores fusion rules and axes in non-associative algebras, establishing dimension bounds and classifying small cases, thereby advancing understanding of algebraic structures related to axial algebras and vertex operator algebras.

Contribution

It introduces new results on the dimension constraints of algebras generated by axes and classifies low-dimensional cases, extending the theory of fusion rules in non-associative algebras.

Findings

Dimension of algebra generated by two axes is at most 5.

Algebras with dimension ≤ 3 are classified up to isomorphism.

Dimension 4 is impossible for such generated algebras.

Abstract

"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are closely related to $3$-transposition groups and Vertex operator algebras. In this paper we consider fusion rules for semisimple idempotents, following Albert in the power-associative case. We examine the notion of an axis in the non-commutative setting and show that the dimension $d$ of any algebra $A$ generated by a pair $a,b$ of (not necessarily Jordan) axes of respective types $(\lambda,\delta)$ and $(\lambda',\delta')$ must be at most $5$; $d$ cannot be $4.$ If $d\le 3$ we list all the possibilities for $A$ up to isomorphism. We prove a variety of additional results and mention some research questions at the end.