Generalised dihedral subalgebras from the Monster

TL;DR

This paper generalizes the classification of certain subalgebras related to the Monster group, exploring new parameter ranges and identifying the largest axial algebras with key properties similar to those associated with the Monster.

Contribution

It extends the known classification by generalizing parameters and constructing larger axial algebras that retain key features of the Monster's subalgebras.

Findings

Identified the largest nonassociative axial algebras satisfying key properties

Established the significance of parameters  and 2 in these algebras

Demonstrated the algebras admit an associating symmetric bilinear form

Abstract

The conjugacy classes of the Monster which occur in the McKay observation correspond to the isomorphism types of certain 2-generated subalgebras of the Griess algebra. Sakuma, Ivanov and others showed that these subalgebras match the classification of vertex algebras generated by two Ising conformal vectors, or of Majorana algebras generated by two axes. In both cases, the eigenvalues parametrising the theory are fixed to $\alpha=\frac14,\beta=\frac1{32}$. We generalise these parameters and the algebras which depend on them, in particular finding the largest possible (nonassociative) axial algebras which satisfy the same key features, by working extensively with the underlying rings. The resulting algebras admit an associating symmetric bilinear form and satisfy the same 6-transposition property as the Monster; $\alpha=\frac14,\beta=\frac1{32}$ turns out to be distinguished.