Quotients of the Highwater algebra and its cover

Primitive axial algebras of Monster type are a class of non -associative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2-generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinite-dimensional Highwater algebra 7-t, or its characteristic 5 cover 7-t.In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of 7-t and 7-t. We proceed in a unified way, by defining a cover of 7-t in all characteristics. This cover 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.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).