A Natural Representation of the FischerGriess Monster with the Modular Function J as Character
Abstract
We announce the construction of an irreducible graded module V for an ``affine'' commutative nonassociative algebra hat{B}. This algebra is an ``affinization'' of a slight variant B 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 + 196884q +.... We obtain a natural action of the Monster on V compatible with the action of hat{B}, 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 A^{1^{()} 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 infinitedimensional representation.
 Publication:

Proceedings of the National Academy of Science
 Pub Date:
 May 1984
 DOI:
 10.1073/pnas.81.10.3256
 Bibcode:
 1984PNAS...81.3256F