In this article, we will prove that the subsectors of $\alpha$-induced sectors for $M \rtimes \hat{ G} \supset M$ forms a modular category, where $M \rtimes \hat{ G}$ is the crossed product of $M$ by the group dual $\hat{ G}$ of a finite group $G$. In fact, we will prove that it is equivalent to M\"uger's crossed product. By using this identification, we will exhibit an orbifold aspect of the quantum double of $\Delta$(not necessarily non-degenerate) obtained from a Longo-Rehren inclusion $A \supset B_\Delta$ under certain assumptions. We will apply the above description of the quantum double of $\Delta$ to the Reshetikhin-Turaev topological invariant of closed 3-manifolds, and we obtain a simpler formula, which is a degenerate version of Turaev's theorem that the Reshetikhin-Turaev invariant for the quantum double of a modular category $\hat{ \Delta}$ is the product of Reshetikhin-Turaev invariant of $\hat{ \Delta}$ and its complex conjugate.
Categories: math.OA math.GT