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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Classifying quotients of the Highwater algebra

## Classifying quotients of the Highwater algebraAdd to your list(s) Download to your calendar using vCal - Justin McInroy (University of Chester)
- Wednesday 27 July 2022, 10:00-10:30
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact nobody. GR2W02 - Simple groups, representations and applications 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 This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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