Classifying quotients of the Highwater algebra

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 $(\alpha, \beta)$ 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 will begin with a reminder of axial algebras which were introduced last week.

This is joint work with: Clara Franchi, Catholic University of the Sacred Heart, Milan Mario Mainardis, University of Udine

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar