ON AXIOMATIC APPROACHES TO VERTEX OPERATOR-ALGEBRAS AND MODULES

The basic definitions and properties of vertex operator algebras, modules, intertwining operators and related concepts are presented, following a fundamental 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 associativity 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.