VERTEX OPERATOR ALGEBRAS AND MODULES 7

vertex operator algebra and some consequences, including the 83-symmetry of

the Jacobi identity. We introduce the standard kinds of elementary categorical

notions appropriate to vertex operator algebras, including tensor products. We

discuss conformal and projective transformations of vertex operators and we de-

fine and comment on the more general class of quasi-vertex operator algebras. In

Section 3 we formulate and prove the basic duality properties of vertex operator

algebras, namely, the rationality of the two kinds of compositions of vertex op-

erators - products and iterations - and commutativity and associativity. Then

we show that these rationality, commutativity and associativity properties may

be used in place of the Jacobi identity in the definition of vertex operator alge-

bras. Our treatment over the field F is based on a formalism of formal Laurent

series expansions of rational functions. We show that the associativity follows

from commutativity for products of two vertex operators together with certain

other properties of vertex operator algebras. We extend rationality and commu-

tativity to several variables, using the Jacobi identity; associativity could also

be extended analogously to several variables. Finally, we apply the duality for-

mulation of vertex operator algebras to prove that the tensor product of vertex

operator algebras is again a vertex operator algebra (a result stated in Section

2).

In Section 4 we define modules for vertex operator algebras and present conse-

quences of the definition, elementary categorical notions and duality properties,

in parallel with the corresponding parts of Section 2. We introduce tensor prod-

uct modules for tensor product algebras and prove that their irreducibility is

equivalent to the irreducibility of the factor modules. We also establish that

under a natural hypothesis, every irreducible module for a tensor product of

algebras is in fact a tensor product of irreducible modules for the factors. In

Section 5 we study the duality properties of modules in greater depth. First we

motivate and construct the vertex operators which correspond to elements of the

module and map from the vertex operator algebra to the module and then we

prove the various forms of duality for one module element and two algebra ele-

ments. We also establish several converse statements which allow us to replace

the Jacobi identity in the definition of module by any of various duality assump-

tions. We then define the notion of adjoint vertex operator (using a formula in

[B]), acting on the suitably restricted dual space of a given module, and using

this, together with appropriate sl(2) relations, we prove that every module has

a natural contragredient module. We show that these contragredient modules

behave similarly to those in classical algebraic theories; in particular, the double

contragredient module is canonically isomorphic to the original module, and irre-

ducibility is preserved under contragredience. As in classical algebraic theories,

we interpret nondegenerate "invariant" bilinear forms and pairings in terms of

contragredient modules. Then we show that if a vertex operator algebra is self-

contragredient as a module for itself, then the corresponding form is symmetric.

Next we define intertwining operators and prove their basic properties. (Our def-

inition is slightly different from that in [MS].) We introduce adjoint intertwining

operators, prove that they are indeed intertwining operators, and give their basic

properties. One important corollary is an 83-symmetry involving the dimensions

of spaces of intertwining operators. These dimensions are termed fusion rules,