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A natural representation of the Fischer-Griess monster with the modular function J as character. (English) Zbl 0543.20016

Summary: We announce the construction of an irreducible graded module V for an ”affine” commutative nonassociative algebra \(\hat {\mathcal B}\). This algebra is an ”affinization” of a slight variant \({\mathcal B}\) 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 \(\hat {\mathcal B}\), 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 \(A_ 1^{(1)}\) and the relation between them play an important role in our construction. As a corollary we deduce Griess’ 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.

MSC:

20D08 Simple groups: sporadic groups
17B65 Infinite-dimensional Lie (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C15 Ordinary representations and characters
17B70 Graded Lie (super)algebras
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