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We announce the construction of an irreducible graded moduleV for an “affine” commutative nonassociative algebra [unk]. This algebra is an “affinization” of a slight variant [unk] of the commutative nonassociative algebraB defined by Griess in his construction of the Monster sporadic groupF 1 . The character ofV is given by the modular functionJ (q ) =q -1 + 0 + 196884q +.... We obtain a natural action of the Monster onV 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 algebraA 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 ofB and the action ofF 1 on it. In this work, the Monster, a finite group, is defined and studied by means of a canonical infinite-dimensional representation.

A natural representation of the Fischer-Griess Monster with the modular function J as character