Introduction

Vertex operators appeared in the mathematics literature in 1978 [9], when

a representation of the affine Kac-Moody Lie algebra A[' — a central exten-

sion of the tensor product of s/(2, C) and the algebra of Laurent polynomials

in one variable — was constructed by means of differential operators now

understood as examples of Z2-twisted vertex operators. Previously, vertex

operators had been used by physicists in dual resonance models.

Within a few years, vertex operators formed a solid bridge between math-

ematics and physics. New constructions and results concerning finite groups,

Lie algebras and modular functions, and new models in string theory, quan-

tum field theory and statistical mechanics have contributed to the establish-

ment of the representation theory of vertex operator algebras — the algebraic

counterpart of two-dimensional conformal quantum field theory — as a cru-

cial area of research. The prototypical role of A[ ' and of its finite-dimensional

subalgebra 5/(2, C) in the study of more complicated symmetries is now un-

derstood in terms of vertex operator algebras and modules obtained from

root lattices of Lie algebras. Suitable vertex operator algebras carry natural

symmetries of two-dimensional conformal field theories and of the Monster

sporadic finite simple group.

In this spirit, this paper is a study of the main axiom for a vertex operator

algebra — the Jacobi identity (for vertex operators) — and its relation with

the standard Ax -modules. We extend the Jacobi identity for a vertex op-

erator algebra over a field of characteristic zero to multi-operator identities.

Next, based on constructions of C. Y. Dong and J. Lepowsky ([2] — [4]), we

establish a Jacobi identity for relative Z2-twisted vertex operators. Then in

the case of A[ , we naturally recover the twisted Z-operators and correspond-

ing generating function identities developed earlier by J. Lepowsky and R.

L. Wilson ([10] — [12]) for their construction of the standard At -modules.

This is accomplished by interpreting these identities as expansion coefficients

of multi-operator extensions of the relative Z2-twisted Jacobi identity.

The present work, which was stimulated by the book "Vertex Operator

Algebras and the Monster" by I. Frenkel, J. Lepowsky and A. Meurman [6]

and by a paper of R. E. Borcherds [1], is deeply rooted in Chapters 8 and

9 of [6] and in the work of C. Y. Dong and J. Lepowsky on relative vertex

operators, Z-algebras and parafermion algebras ([2] — [4]).

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