Introduction
0.1.The remarkable link between the structures of mathematics and physics is exemplified by numerous instances in the development of these two disciplines. This link has remained ever mysterious. It comes up in the most unexpected situations and only further developments shed light on the meaning of that which is hidden behind the formal coincidence. The present article gives one more example of such a link between a mathematical and a physical theory.Our original goal was to construct explicitly the simplest non-trivial highest weight representation (the so-called basic representation) of a Kac-Moody Lie algebra with the affine Cartan matrix. This seemed to be possible to do for two reasons. First, an explicit character formula is known [6] and second, there exists a construction of this representation [8], which is based on the specialization of the general character formula from [5]. Unfortunately the latter construction is very "inhomogeneous" (and has nothing to do with the explicit character formula). For a long time each of the authors had been trying unsuccessfully to attack the problem until they turned to physics where the answer had been found. It turned out that the basic representation can be described in terms of the vertex operators which play a crucial role in the dual resonance theory! H. Garland noticed the similarity between the operators in the "non-homogeneous" construction of the basic representation and the vertex operators. But that was only a similarity. In our construction the coincidence is complete, and this allows us to include the dual resonance models in the framework of the theory of affine Lie algebras. We hope that the mathematical instrument of affine Lie algebras will have useful physical applications, thereby repaying the debt.Affine Lie algebras (which are alternatively known as Kac-Moody Lie algebras with an affine Cartan matrix, and also as loop algebras) form an important class of infinite dimensional Lie algebras. These algebras together 24 I.B. Frenkel and V.G. Kac with the Lie algebra of vector fields on the circle essentially exhaust all infinitedimensional Lie algebras which admit a 7/-gradation by subspaces of bounded dimension and have no graded ideals [4]. The structure of affine Lie algebras is similar to that of simple finite dimensional Lie algebras, which permit to generalize many results of the classical theory [4], [11], [2]. On the other hand, the quotient of an affine Lie algebra by a 1-dimensional center is isomorphic to the Lie algebra of polynomial maps from the circle into a simple Lie algebra. This allows one to study the affine Lie algebras from a different point of view. In particular, they admit an extension by a Lie algebra of vector fields on the circle. As a result the Virasoro algebra, which is a central extension of the latter Lie algebra, operates on the space of a highest weight representation. This gives some information about representations of the Virasoro algebra [7], which are important in physical applications,
0.2,It is in...
We announce the construction of an irreducible graded module V for an "affine" commutative nonassociative algebra S. This algebra is an "affinization" of a slight variant R of the conMnutative nonassociative algebra B defined by Griess in his construction of the Monster sporadic group F1. The character of V is given by the modular function J(q) _ q' + 0 + 196884q + .... We obtain a natural action of the Monster on V compatible with the action of R, 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 cadculus of vertex operators. In particular, the homogeneous and principal representations of the simplest affine Lie algebra AV') 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 F1 on it. In this work, the Monster, a finite group, is defined and studied by means of a canonical infinite-dimensional representation.
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