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 $V^{\chi }$ 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_χ$isabijectionfromthesetofirreducible$G$-modulestothesetof(inequivalent)simple$V^G$-moduleswhicharecontainedin$V.$

25 pages, latex, no figures

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