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, or its characteristic 5 cover. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of the Highwater algebra and its characteristic 5 cover. We proceed in a unified way, by defining a cover of the Highwater algebra 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.