Quotients of the Highwater algebra and its cover

TL;DR

This paper classifies quotients of the Highwater algebra and its cover, completing the understanding of symmetric 2-generated primitive axial algebras of Monster type, and introduces a new algebra with a unique fusion law.

Contribution

It classifies the ideals of the Highwater algebra and its cover, providing explicit bases and a unified approach across all characteristics, revealing new algebraic structures.

Findings

Classified ideals of the Highwater algebra and its cover.

Provided explicit bases for these ideals.

Discovered a new algebra with a unique fusion law.

Abstract

Axial algebras are a class of non-associative algebra with a strong link to finite (especially simple) groups which have recently received much attention. Of primary interest are the axial algebras of Monster type $(\alpha, \beta)$, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. In this paper, we complete the classification of the symmetric $2$-generated primitive axial algebras of Monster type $(\alpha, \beta)$. By previous work of Yabe, and Franchi and Mainardis, any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra $\mathcal{H}$, or its characteristic $5$ cover $\hat{\mathcal{H}}$. In this paper, we classify the ideals of $\mathcal{H}$ and $\hat{\mathcal{H}}$ and thus their quotients. Moreover, we give explicit bases for the ideals. In fact, we proceed in a unified way, by defining a cover $\hat{\mathcal{H}}$ of $\mathcal{H}$ in all characteristics and classifying its ideals. Our new algebra $\hat{\mathcal{H}}$ has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic $5$.