Vertex operator algebras and Kazhdan-Lusztig's tensor category

The main goal of this thesis is to study and illuminate, from the viewpoint of vertex operator algebras, a certain braided tensor category of Kazhdan and Lusztig based on certain modules for an affine Lie algebra. We use a new logarithmic generalization, due to Huang, Lepowsky and Zhang, of Huang and Lepowsky's tensor product theory for modules for a vertex operator algebra. We also apply this generalized tensor product theory to vertex algebras based on lattices.; We first motivate the tensor product theory and introduce some basic notions and results. Then we study the "compatibility condition," which originated as an important tool in the tensor product theory of Huang and Lepowsky. We give a "strong lower truncation condition" and prove its equivalence to the usual lower truncation together with the compatibility condition. We then focus on the category considered by Kazhdan and Lusztig and build the vertex operator algebraic setup for a tensor product theory for this category. Using the new truncation condition we prove the equivalence of the tensor product functor constructed by Kazhdan and Lusztig and the one constructed in the logarithmic tensor product theory. We use certain generalized Knizhnik-Zamolodchikov equations to prove the "convergence and expansion properties" for this category and obtain a new construction of the natural associativity isomorphisms, and hence of the braided tensor category structure. In addition, from this we obtain vertex tensor category structure on this category.; Next we study the module category for a "conformal vertex algebra" constructed from a hyperbolic even lattice. We carry out the tensor product theory for such a category and study the resulting vertex tensor categories and braided tensor categories. At the end we prove a result, also related to tensor product theory, concerning the relation between the lower truncation condition and the "Jacobi identity."...