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
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
(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.