REGULAR REPRESENTATIONS OF VERTEX OPERATOR ALGEBRAS

Abstract

This paper is to establish a theory of regular representations for vertex operator algebras. In the paper, for a vertex operator algebra V and a V-module W, we construct, out of the dual space W*, a family of canonical weak V ⊗ V-modules [Formula: see text] with a nonzero complex number z as the parameter. We prove that for V-modules W, W 1 and W 2 , a P(z)-intertwining map of type [Formula: see text] in the sense of Huang and Lepowsky exactly amounts to a V ⊗ V-homomorphism from W 1 ⊗ W 2 into [Formula: see text]. Combining this with Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of P(z)-intertwining maps of the same type we obtain a canonical linear isomorphism fromthe space [Formula: see text] of intertwining operators of the indicated type to [Formula: see text]. Denote by R P(z) (W) the sum of all (ordinary) V ⊗ V-submodules of [Formula: see text]. Assuming that V satisfies certain suitable conditions, we obtain a canonical decomposition of R P(z) (W) into irreducible V ⊗ V-modules. In particular, we obtain a decomposition of Peter–Weyl type for R P(z) (V). Denote by ℱ P(z) the functor from the category of V-modules to the category of weak V ⊗ V-modules such that ℱ P(z) (W)=R P(z) (W'). We prove that for V-modules W 1 , W 2 , a P(z)-tensor product of W 1 and W 2 in the sense of Huang and Lepowsky exactly amounts to a universal from W 1 ⊗ W 2 to the functor ℱ P(z) . This implies that the functor ℱ P(z) is essentially a right adjoint of the Huang–Lepowsky's P(z)-tensor product functor. It is also proved that R P(z) (W) for [Formula: see text] are canonically isomorphic V ⊗ V-modules.