# Manchester Algebra Seminar - Justin Mcinroy

Dates: | 9 November 2021 |
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Times: | 13:00 - 14:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

Speaker: Justin Mcinroy - Bristol

Title: Classifying quotients of the Highwater algebra

Abstract: Axial algebras are a new class of non-associative algebras which have a strong link to groups. They are generated by axes which are semisimple idempotents whose eigenvectors multiply according to a so-called fusion law. The prototypical example is the Griess algebra whose automorphism group is the Monster. The eigenvalues for the axes in the Griess algebra are 1, 0, 1/4 and 1/32 and we call the fusion law Monster type M(1/4, 1/32).

Recently Yabe has classified all the symmetric 2-generated axial algebras with the generalised Monster fusion law $\mathcal{M}(\alpha, \beta)$, where $\alpha$ and $\beta$ are indeterminants. He showed they either belong to a list of families of algebras, or are a quotient of the infinite-dimensional Highwater algebra, or its characteristic 5 cover. We classify all the ideals of the Highwater algebra (and its cover) and hence make Yabe’s classification explicit. 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 but 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, Milan Mario Mainardis, University of Udine

Place: Frank Adams (and to be streamed online*)

- subject to equipment and connection

Tea and biscuits 12:45 in the foyer