8 IGOR B. FRENKEL, YI-ZHI HUANG AND JAMES LEPOWSKY
as in [V], because of their analogy with the multiplicities of irreducible modules
in tensor products of irreducible modules. We end the section with the formula-
tion and proof of the various forms of duality for two module elements and one
algebra element. This is based on the use of adjoint intertwining operators to
construct vertex operators which correspond to elements of a module and map
from the module to the algebra. The proof of these duality relations uses most
of the concepts and arguments presented throughout the paper. This concludes
our treatment of the monodromy-free axiomatic foundation of the general theory
of vertex operator algebras, modules and intertwining operators.
Much of this work, which is an outgrowth of the book [FLM], was done during
the final stages of the writing of [FLM]. We would like to thank Robert Wilson for
helpful comments. We gratefully acknowledge partial support from the follow-
ing sources: I.F.: the Institute for Advanced Study, Institut des Hautes Etudes
Scientifiques and National Science Foundation grants DMS-8602091 and DMS-
8906772; Y.Z.H.: a Rutgers University Graduate Excellence Fellowship and a
Sloan Foundation Doctoral Dissertation Fellowship; J.L.: a Guggenheim Foun-
dation Fellowship, the Rutgers University Faculty Academic Study Program and
National Science Foundation grant DMS-8603151.
Additional note: After this work was completed we received the preprint of
the paper [DGM], which includes a new proof and generalization of one of the
main theorems in [FLM]; some of the methods in [DGM] are similar to some of
the material in the present paper.
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