Regular representations and Huang–Lepowsky tensor functors for vertex operator algebras

This is the second paper in a series devoted to studies of regular representations for vertex operator algebras. In this paper, given a module W for a vertex operator algebra V, we construct, from the dual space W*, a family of canonical (weak) V V-modules called parameterized by a nonzero complex number z. We prove that for V-modules W, W1, and W2, a Q(z)-intertwining map of type in the sense of Huang and Lepowsky exactly amounts to a V V-homomorphism from W1 W2 to and that a Q(z)-tensor product of V-modules W1 and W2 in the sense of Huang and Lepowsky amounts to a universal from W1 W2 to the functor , where is a functor from the category of V-modules to the category of weak V V-modules defined by for a V-module W. Furthermore, Huang–Lepowskyʹs P(z)- and Q(z)-tensor functors for the category of V-modules are extended to functors TP(z) and TQ(z) from the category of V V-modules to the category of V-modules. It is proved that functors and are right adjoints of TP(z) and TQ(z), respectively.