Intertwining Operator Algebras and Vertex Tensor Categories for Affine Lie Algebras

0. Introduction. The category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level k for an affine Lie algebra ĝ is particularly important from the viewpoint of conformal field theory and related mathematics. Here we call this the category generated by the standard ĝ-modules of level k. A central theme is a braided tensor category structure (in the sense of Joyal and Street [JS]) on this category, a structure explicitly discovered by Moore and Seiberg [MS] in their important study of conformal field theories. In [MS], Moore and Seiberg constructed this structure based on the assumption that there exists a suitable operator product expansion for chiral vertex operators; this is essentially equivalent to assuming the associativity of intertwining operators, in the language of vertex operator algebra theory. Actually, Belavin, Polyakov, and Zamolodchikov [BPZ] had already formalized the relation between the operator product expansion and representation theory in the context of conformal field theory, especially for the Virasoro algebra, and Knizhnik and Zamolodchikov [KZ] had established the relation between conformal field theory and the representation theory of affine Lie algebras. In [KL1]–[KL5], Kazhdan and Lusztig achieved a breakthrough by indeed constructing a natural braided tensor category structure, with the additional property of rigidity, on a certain category of ĝ-modules of level k, when k is sufficiently negative, or more generally, when k is in a certain large subset of C excluding the positive integers, and particularly, proving that this braided tensor category is equivalent to a tensor category of modules for a quantum group constructed from the same finitedimensional Lie algebra. The method used by Kazhdan and Lusztig, especially in their construction of the associativity isomorphisms, is algebro-geometric and is closely related to the algebro-geometric formulation and study of conformal-field-theoretic structures in the influential works of Tsuchiya, Ueno, and Yamada [TUY], Drinfeld [Dr], and Beilinson, Feigin, and Mazur [BFM]. (The work [BFM] discusses the case of minimal models for the Virasoro algebra, and Beilinson informs us that a similar argument works for the case of the category generated by the standard ĝ-modules.) In the important work [F1] and [F2], Finkelberg proved that the braided tensor