An analogue of the Hom functor and a generalized nuclear democracy theorem

The notion of vertex operator algebra ([B], [FHL], [FLM]) is the algebraic counterpart of the notion of what is now usually called “chiral algebra” in conformal field theory, and vertex operator algebra theory generalizes the theories of affine Lie algebras, the Virasoro algebra and representations (cf. [B], [DL], [FLM], [FZ]). It has been well known (cf. [FZ], [L1]) that the irreducible highest weight modules (usually called the vacuum representations) L(l, 0) for an affine Lie algebra ĝ of level l and L(c, 0) for the Virasoro algebra with central charge c have natural vertex operator algebra structures. If l is a positive integer, it was proved ([DL], [FZ], [L1]) that the category of L(l, 0)-modules is a semi-simple category whose irreducible objects are irreducible highest weight integrable ĝ-modules of level l (cf. [K]). If c = 1 − 6(p−q) 2