Axial algebras for sporadic simple groups HS and Suz.

Abstract

Motivated by the construction of the Monster sporadic simple group as a group of automorphisms of an algebra and the recent development of ax- ial algebras as a generalization of Majorana representations, we construct axial algebras for the sporadic simple groups HS and Suz in different ways analogous to the Norton algebra construction. We study how these algebras decompose as direct sums of the adjoint action of an axis. Fusion rules, that is the rules with which eigenvectors from various eigenspaces of the adjoint action multiply, are found. This places these groups in a general framework of groups acting on algebras hence giving a common theme for their origin.