[Paper Review] Regular representations of vertex operator algebras, I

Haisheng Li|ArXiv.org|Aug 19, 1999

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper constructs a canonical weak $V \otimes V$-module ${\cal{D}}_{P(z)}(W)$ from a module $W$ of a vertex operator algebra $V$ and a complex number $z$, establishing a one-to-one correspondence between intertwining operators of type ${W' \choose W_1 W_2}$ and $V \otimes V$-homomorphisms from $W_1 \otimes W_2$ into ${\cal{D}}_{P(z)}(W)$. The key contribution is a Peter-Weyl-type decomposition of the regular representation ${\cal{D}}_{P(z)}(V)$ when $W = V$, providing a structural framework for intertwining operators in VOAs.

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 $V$.

Motivation & Objective

• To establish a canonical construction of $V \otimes V$-modules from $V$-modules and complex parameters $z$.
• To provide a categorical framework linking intertwining operators to $V \otimes V$-homomorphisms.
• To generalize the theory of intertwining operators using a geometric $P(z)$-channel construction.
• To derive a Peter-Weyl-type decomposition for the regular representation ${\cal{D}}_{P(z)}(V)$ when $W = V$.

Proposed method

• Define the module ${\cal{D}}_{P(z)}(W)$ as a subspace of the dual space $W^*$, parameterized by $z \in \mathbb{C}^\times$.
• Use Huang and Lepowsky's linear correspondence between intertwining operators and $P(z)$-intertwining maps to relate operator spaces to homomorphism spaces.
• Construct a canonical linear isomorphism from the space of intertwining operators ${\cal{V}}^{W'}_{W_1 W_2}$ to $\Hom_{V \otimes V}(W_1 \otimes W_2, {\cal{D}}_{P(z)}(W))$.
• Prove that $P(z)$-intertwining maps of type ${W' \choose W_1 W_2}$ correspond exactly to $V \otimes V$-homomorphisms into ${\cal{D}}_{P(z)}(W)$.
• Analyze the structure of ${\cal{D}}_{P(z)}(V)$ as a regular representation of $V$, leading to a decomposition analogous to Peter-Weyl theory.
• Apply the theory to the case $W = V$ to obtain a decomposition of the regular representation into irreducible components.

Experimental results

Research questions

• RQ1 How can one systematically construct $V \otimes V$-modules from $V$-modules and complex parameters $z$?
• RQ2 What is the precise relationship between $P(z)$-intertwining maps and $V \otimes V$-homomorphisms?
• RQ3 Can the space of intertwining operators be canonically identified with a space of $V \otimes V$-module homomorphisms?
• RQ4 What structural decomposition arises in the regular representation ${\cal{D}}_{P(z)}(V)$?
• RQ5 Does the construction yield a Peter-Weyl-type decomposition for the regular representation of $V$?

Key findings

• A canonical weak $V \otimes V$-module ${\cal{D}}_{P(z)}(W)$ is constructed as a subspace of $W^*$ for any $V$-module $W$ and $z \in \mathbb{C}^\times$.
• There exists a canonical linear isomorphism between the space of intertwining operators of type ${W' \choose W_1 W_2}$ and the space of $V \otimes V$-module homomorphisms from $W_1 \otimes W_2$ to ${\cal{D}}_{P(z)}(W)$.
• The construction establishes a one-to-one correspondence between $P(z)$-intertwining maps of type ${W' \choose W_1 W_2}$ and $V \otimes V$-homomorphisms into ${\cal{D}}_{P(z)}(W)$.
• For $W = V$, the module ${\cal{D}}_{P(z)}(V)$ admits a Peter-Weyl-type decomposition, identifying it as the regular representation of $V$.
• The theory provides a unified framework for classifying intertwining operators via representation-theoretic homomorphisms in $V \otimes V$-modules.
• The results generalize and categorize the structure of intertwining operators in vertex operator algebras using module-theoretic constructions.