A natural representation of the Fischer-Griess Monster with the modular function <i>J</i> as character

TLDR

The algebra studied is an affinization of a variant of Griess’s commutative nonassociative algebra that underlies the construction of the Monster sporadic group. The authors construct an irreducible graded module for an affine commutative nonassociative algebra and use it to define and study the Monster group through a canonical infinite-dimensional representation. The construction employs affine Lie algebra representation theory, vertex operator calculus, and the modular function J as the character of the graded module. The module admits a natural Monster action that conceptually explains much of Monstrous Moonshine, and the construction recovers Griess’s results on the algebra structure of B and the action of F1.

Abstract

We announce the construction of an irreducible graded module V for an “affine” commutative nonassociative algebra [unk]. This algebra is an “affinization” of a slight variant [unk] of the commutative nonassociative algebra B defined by Griess in his construction of the Monster sporadic group F 1 . The character of V is given by the modular function J ( q ) = q -1 + 0 + 196884 q +.... We obtain a natural action of the Monster on V compatible with the action of [unk], thus conceptually explaining a major part of the numerical observations known as Monstrous Moonshine. Our construction starts from ideas in the theory of the basic representations of affine Lie algebras and develops further the calculus of vertex operators. In particular, the homogeneous and principal representations of the simplest affine Lie algebra A 1 (l) and the relation between them play an important role in our construction. As a corollary we deduce Griess's results, obtained previously by direct calculation, about the algebra structure of B and the action of F 1 on it. In this work, the Monster, a finite group, is defined and studied by means of a canonical infinite-dimensional representation.