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]). 1
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