6 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY

state of development of the theory naturally leads to the axiom system chosen

here (and in [PLM]).

It is by no means obvious that nontrivial examples of vertex operator algebras

exist. In fact, the construction of such objects is quite a long story, not treated in

this paper, which is devoted instead to axiomatic considerations. For instance,

the construction of the moonshine module for the Monster, together with related

vertex operator algebras and modules, is presented in detail in [FLM].

As we have been discussing, vertex operator algebras are similar in spirit to

Lie algebras. However, they are also similar in spirit to commutative associative

algebras: The Jacobi identity is equivalent to two forms of "duality" (in the

language of conformal field theory):

y(ti, z)Y(v, z0) ~ Y(v, zo)Y(u, z) (1.18)

Y(u, z)Y(v, zo) ~ Y(Y(u, z - z0)v, *), (1.19)

where the symbol ~ is understood as an equality of the (operator-valued) ratio-

nal functions corresponding to the formal Laurent series defined by the indicated

expressions. In fact, the function F in (1.13) is taken to be an arbitrary matrix

coefficient of this common rational function (cf. (1.11)). Moreover, the second

form of duality (1.19) follows from the first (1.18) and properties of L(—1) and

L(0), and this fact provides a useful approach to constructing vertex operator al-

gebras. The second form of duality is known as the associativity of the operator

product expansion in conformal field theory. Although vertex operator algebras

are not actually associative algebras, such results as the existence of tensor prod-

ucts of vertex operator algebras and the factorizability of irreducible modules for

such tensor products still hold in the vertex operator algebra context. On the

other hand, an additional feature of the theory (besides the Jacobi identity)

analogous to Lie algebra theory as opposed to associative algebra theory is the

concept of contragredient module and the associated adjoint operators. These

concepts involve the nontrivial use of si(2). It is also interesting that it is some-

times valuable, within a number of arguments, to pass back and forth between

the equivalent axiom systems. For instance, to prove the Jacobi identity for a

tensor product of vertex operator algebras, it is easiest to verify the duality rela-

tions instead, and to prove the appropriate general duality relations for modules,

it is easiest to use the ^-symmetr y of the Jacobi identity. As another example,

the Jacobi identity is useful in extending commutativity to products of several

vertex operators. The feature of vertex operator algebras that on the one hand

they are very similar to the classical structures of Lie and associative algebras,

and that on the other hand they possess enormously rich structure embracing

many previously unrelated classical objects, as referred to above, is one of the

most beautiful aspects of this new theory.

This paper is devoted to the basic properties of vertex operator algebras, their

representations and their intertwining operators that we have mentioned. It is

written in an essentially self-contained way, although a brief look at Chapters 2

and 8 of [FLM] might be helpful to the reader. We work in an algebraic setting

over the general field F of characteristic 0. In Section 2 we give the definition of