Regular representations of vertex operator algebras, I

TL;DR

This paper constructs a canonical $V\otimes V$-module from a $V$-module and a complex parameter, establishing a correspondence with intertwining maps and decomposing regular representations akin to Peter-Weyl theory.

Contribution

It introduces a new construction of regular representations of vertex operator algebras and links intertwining operators to $V\otimes V$-homomorphisms, expanding the understanding of module interactions.

Findings

Established a canonical $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ from a $V$-module $W$.

Proved a correspondence between $P(z)$-intertwining maps and $V\otimes V$-homomorphisms.

Decomposed ${\cal{D}}_{P(z)}(V)$ into a Peter-Weyl type sum for regular representations.

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 ${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$ depending on $z$). We prove that for $V$-modules $W, W_{1}$ and $W_{2}$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ ([H3], [HL0-3]) exactly amounts to a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. 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 ${\cal{V}}^{W'}_{W_{1}W_{2}}$ 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 ${\cal{D}}_{P(z)}(V)$, which are what we call the regular representations of $V$.