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Enumerating 3-generated axial algebras of Monster type
McInroy, Justin Shpectorov, Sergey Khasraw, Sanhan
McInroy, Justin
Shpectorov, Sergey
Khasraw, Sanhan
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2021-06-17
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Abstract
An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra.
The 2-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of 3-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.
Citation
Khasraw, S. M. S., McInroy, J., & Shpectorov, S. (2022). Enumerating 3-generated axial algebras of Monster type. Journal of Pure and Applied Algebra, 226(2), 106816. https://doi.org/10.1016/j.jpaa.2021.106816
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Elsevier
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Journal of Pure and Applied Algebra
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DOI
10.1016/j.jpaa.2021.106816
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0022-4049