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

Abstract

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 D Q(z) (W) parameterized by a nonzero complex number z . We prove that for V -modules W , W 1 , and W 2 , a Q ( z )-intertwining map of type W′ W 1 W 2 in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V -homomorphism from W 1 ⊗ W 2 to D Q(z) (W) and that a Q ( z )-tensor product of V -modules W 1 and W 2 in the sense of Huang and Lepowsky amounts to a universal from W 1 ⊗ W 2 to the functor F Q(z) , where F Q(z) is a functor from the category of V -modules to the category of weak V ⊗ V -modules defined by F Q(z) (W)= D Q(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 T P ( z ) and T Q ( z ) from the category of V ⊗ V -modules to the category of V -modules. It is proved that functors F P(z) and F Q(z) are right adjoints of T P ( z ) and T Q ( z ) , respectively.