From forbidden configurations to a classification of some axial algebras of Monster type

Justin McInroy; Sergey Shpectorov

doi:10.48550/arXiv.2107.07415

From forbidden configurations to a classification of some axial algebras of Monster type

Justin McInroy, Sergey Shpectorov

Mathematics

Research output: Working paper/Preprint › Preprint

Abstract

Ivanov introduced the shape of a Majorana algebra as a record of the $2$-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper.

Original language	English
Publisher	arXiv
Pages	1-48
Number of pages	48
DOIs	https://doi.org/10.48550/arXiv.2107.07415
Publication status	Published - 30 Dec 2022

Bibliographical note

In this version we have updated the sections on axets and shapes. We have changed language from isogenies to the more categorical language of morphisms, which has simplified the exposition. We have also updated Subsection 7.4 on quotients of the Highwater algebra, following the new notation in the preprint of Franchi, Mainardis and McInroy. 47 pages

Keywords

math.RA
math.GR
17A36, 17A60, 20B25, 20F29

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abstract = "Ivanov introduced the shape of a Majorana algebra as a record of the $2$-generated subalgebras arising in that algebra. As a broad generalisation of this concept and to free it from the ambient algebra, we introduce the concept of an axet and shapes on an axet. A shape can be viewed as an algebra version of a group amalgam. Just like an amalgam, a shape leads to a unique algebra completion which may be non-trivial or it may collapse. Then for a natural family of shapes of generalised Monster type we classify all completion algebras and discover that a great majority of them collapse, confirming the observations made in an earlier paper.",

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From forbidden configurations to a classification of some axial algebras of Monster type

Abstract

Bibliographical note

Keywords

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