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

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.

arXiv commentIn this version, we have edited the section on axets, introducing the factor axet and versions of the First Isomorphism Theorem and Correspondence Theorem for axets