1. I N T R O D U C T I O N

Vertex operator algebras are a new and fundamental class of algebraic struc-

tures which have recently arisen in mathematics and physics. Their definition

and some elementary properties and basic examples were presented in the book

[FLM]. The goal of this paper is to lay some further foundations of the theory

of vertex operator algebras and their representations.

The importance of these new algebras is supported by their numerous relations

with other directions in mathematics and physics, such as the representation

theory of the Virasoro algebra and affine Lie algebras, the theory of Riemann

surfaces, knot invariants and invariants of three-dimensional manifolds, quantum

groups, monodromy associated with differential equations, and conformal and

topological field theories. In fact, the theory of vertex operator algebras can be

thought of as an algebraic foundation of a great number of constructions in these

theories.

The main original motivation for the introduction of the notion of vertex op-

erator algebra arose from the problem of realizing the Monster sporadic group

as a symmetry group of a certain infinite-dimensional graded vector space with

natural additional structure. (See the Introduction in [FLM] for a historical

discussion, including the important role of Borcherds' announcement [B].) The

additional structure can be expressed in terms of the axioms defining these new

algebraic objects (which are not actually algebras, even nonassociative algebras,

in the usual sense). The Monster is in fact the symmetry group of a special

vertex operator algebra, the moonshine module, just as the Mathieu group M24

is the symmetry group of a special error-correcting code, the Golay code, and

the Conway group Coo is the symmetry group of a special positive definite even

lattice, the Leech lattice. All three special objects possess and can be character-

ized by the following properties (the uniqueness being conjectural in the Monster

case):

(a) "self-dual"

(b) "rank 24"

(c) "no small elements,"

which have appropriate definitions for each of the three types of mathematical

structures. In the case of vertex operator algebras the notion of self-duality

means that there is only one irreducible module (the moonshine module itself).

Thus even apart from other concepts in mathematics and physics, the Monster

alone leads to the notions of vertex operator algebras and their representations.

Received by editor September 2, 1991.

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