Majorana algebras generated by a 2A algebra and one further axis

TL;DR

This paper classifies Majorana algebras generated by specific axes, linking them to subgroups of the Monster group, and shows that only certain groups admit such representations, extending understanding of algebraic structures related to the Monster.

Contribution

It identifies which groups can produce Majorana algebras generated by a 2A algebra and an additional axis, connecting algebraic and group-theoretic classifications.

Findings

27 groups admit Majorana representations

These groups are subgroups of the Monster generated by three 2A-involutions

Certain groups do not admit Majorana representations

Abstract

We consider Majorana algebras generated by three Majorana axes $a_0$, $a_1$ and $a_2$ such that $a_0$ and $a_1$ generate a dihedral algebra of type 2A. We show that such an algebra must occur as a Majorana representation of one of 27 groups. These 27 groups coincide with the subgroups of the Monster which are generated by three 2A-involutions $a$, $b$ and $c$ such that $ab$ is also a 2A-involution, which were classified by S. P. Norton in 1985. Our work relies on that of S. Decelle and consists of showing that certain groups do not admit Majorana representations.