A B S T R A C T

The basic definitions and properties of vertex operator algebras, mod-

ules, intertwining operators and related concepts are presented, following a fun-

damental analogy with Lie algebra theory. The first steps in the development

of the general theory are taken, and various natural and useful reformulations

of the axioms are given. In particular, it is shown that the Jacobi(-Cauchy)

identity for vertex operator algebras - the main axiom - is equivalent (in the

presence of more elementary axioms) to rationality, commutativity and associa-

tivity properties of vertex operators, and in addition, that commutativity implies

associativity. These "duality" properties and related properties of modules are

crucial in the axiomatic formulation of conformal field theory. Tensor product

modules for tensor products of vertex operator algebras are considered, and it is

proved that under appropriate hypotheses, every irreducible module for a tensor

product algebra decomposes as the tensor product of (irreducible) modules. The

notion of contragredient module is formulated, and it is shown that every module

has a natural contragredient with certain basic properties. Adjoint intertwining

operators are defined and studied. Finally, most of the ideas developed here are

used to establish "duality" results involving two module elements, in a natural

setting involving a module with integral weights.

Key words and phrases. Vertex operator algebras, Jacobi(-Cauchy) identity, Virasoro al-

gebra, duality for vertex operator algebras, modules for vertex operator algebras, intertwining

operators for vertex operator algebras, conformal field theory.