Regular representations of vertex operator algebras, I

Abstract

In this paper, given a module $W$ for a vertex operator algebra $V$ and a nonzero complex number $z$ we construct a canonical (weak) $V\otimes V$-module ${\mathcal{D}}_{P(z)}(W)$ (a subspace of $W^{\ast }$ depending on $z$). We prove that for $V$-modules $W,W_1$ and $W_2$, a $P(z)$-intertwining map of type $\left(\genfrac{}{}{0ex}{}{W^{\prime }}{W_1{W}_2}\right)$ ([H3], [HL0-3]) exactly amounts to a $V\otimes V$-homomorphism from $W_1\otimes W_2$ into ${\mathcal{D}}_{P(z)}(W)$. Using 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 from the space ${\mathcal{V}}_{W_1{W}_2}^{W^{\prime }}$ of intertwining operators of the indicated type to $\Hom_{V\otimes V}(W_{1}\otimes W_{2},{\cal{D}}_{P(z)}(W))$. In the case that $W=V$, we obtain a decomposition of Peter-Weyl type for ${\mathcal{D}}_{P(z)}(V)$, which are what we call the regular representations of $V$.

42 pages, latex