On quantum Galois theory

Abstract

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