Abstract

Axial algebras are a class of non-associative algebras with a strong natural link to groups and have recently received much attention.  They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law.  Of primary interest are the axial algebras with the Monster type $(\alpha, \beta)$ fusion law, of which the Griess algebra (with the Monster as its automorphism group) is an important motivating example. By previous work of Yabe, and Franchi and Mainardis, any symmetric 2-generated axial algebra of Monster type $(\alpha, \beta)$ is either in one of several explicitly known families, or is a quotient of the infinite-dimensional Highwater algebra $\mathcal{H}$, or its characteristic 5 cover $\hat{\mathcal{H}}$.  We complete this classification by explicitly describing the infinitely many ideals and thus quotients of the Highwater algebra (and its cover).  As a consequence, we find that there exist 2-generated algebras of Monster type $\mathcal{M}(\alpha, \beta)$ with any number of axes (rather than just $1,2,3,4,5,6, \infty$ as we knew before) and of arbitrarily large finite dimension. In this talk, we do not assume any knowledge of axial algebras. This is joint work with:Clara Franchi, Catholic University of the Sacred Heart, MilanMario Mainardis, University of Udine