Vertex tensor category structure on a category of Kazhdan-Lusztig

摘要

We incorporate a category considered by Kazhdan andLusztig of certain modules (of a fixed level ℓ, not a positive integer) for an affine Lie algebra, into the representation theory of vertex operator algebras. We do this usingthe logarithmic tensor product theory for generalized modules for a vertexoperator algebra developed by Huang,Lepowsky and the author; we prove that the conditions forapplying this general logarithmic tensor product theory hold. As a consequence, we prove that this categoryhas a natural vertex tensor category structure, and in particularwe obtain a new, vertex-algebraic, construction of the natural associativity isomorphisms and proof of their properties.