On quantum Galois theory

TL;DR

This paper establishes a quantum Galois correspondence for simple vertex operator algebras with finite automorphism groups, showing how the algebra decomposes into modules related to the group's irreducible characters.

Contribution

It proves the non-vanishing of each isotypic component and describes their structure as tensor products, linking group representations to modules of the fixed point subalgebra.

Findings

Each $V^{ ext{ extit{chi}}}$ is nonzero.

$V^{ ext{ extit{chi}}}$ decomposes as $M_{ ext{ extit{chi}}} ensor V_{ ext{ extit{chi}}}$.

Bijection between irreducible $G$-modules and simple $V^G$-modules when $G$ is solvable.

Abstract

For a simple vertex operator algebra $V$ and a finite automorphism group $G$ of $V$ then $V$ is a direct sum of $V^{\chi}$ where $\chi$ are irreducible character of $G$ and $V^{\chi}$ is the subspace of $V$ which $G$ acts according to the character $\chi.$ We prove the following: 1. Each $V^{\chi}$ is nonzero. 2. $V^{\chi}$ is a tensor product $M_{\chi}\otimes V_{\chi}$ where $M_{\chi}$ is an irreducible $G$-module affording $\chi$ and $V_{\chi}$ is a $V^G$-module. If $G$ is solvable, $V_{\chi}$ is a simple $V^G$-module and $M_{\chi}\mapsto $V_{\chi}$ is a bijection from the set of irreducible $G$-modules to the set of (inequivalent) simple $V^G$-modules which are contained in $V.$