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

This is the second paper in a series to study regular representations for vertex operator algebras. In this paper, given a module W for a vertex operator algebra V , we construct, out of the dual space W , a family of canonical (weak) V ⊗V -modules called DQ(z)(W ) parametrized by a nonzero complex number z. We prove that for V -modules W,W1 and W2, a Q(z)-intertwining map of type ( W ′ W1W2 ) in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V -homomorphism from W1 ⊗ W2 to DQ(z)(W ) 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 FQ(z), where FQ(z) is a functor from the category of V -modules to the category of weak V ⊗V -modules defined by FQ(z)(W ) = DQ(z)(W ) for a V -module W . Furthermore, Huang-Lepowsky’s 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 FP (z) and FQ(z) are right adjoints of TP (z) and TQ(z), respectively.