canonical and abstract. In fact, one can define $A_{\alpha}(G,\S)$ for any finite group $G,$ any finite right $G$-set $\S$ and a suitable 2-cocycle on $G$ (see Section 3). This algebra is essentially the crossed product in the theory of Hopf algebra (see \[S\], \[BCM\] and \[DT\]). It turns out that $A_{\alpha}(G,\S)$ is the right algebra in the study of general orbifold theory. In the case that $\S$ is the dual basis of $\C[G]^*,$ $A_{\alpha}(G,\S)$ is exactly the twisted double $D_{\alpha}(G)$ introduced in \[M\] and \[DM5\] in the study of holomorphic orbifold theory. The twisted double $D_{\alpha}(G)$ is conjecturally isomorphic to the twisted quantum double introduced in \[DPR\] and \[D\].

A main tool in the proof of the main theorems is a series of associative algebras $A_n(V)$ constructed in \[DLM4\] for nonnegative integers $n.$ The original motivation for introducing and studying the $A_n(V)$ comes from the representation theory of vertex operator algebras. Let $M=\sum_{n\geq 0}M(n)$ be an admissible $V$-module with $M(0)\ne 0$ (see Section 4 for the definition of admissible module). The algebra $A_n(V)$ is a suitable quotient of $V$ and “takes care” of the first $n+1$ pieces of $M$: each $M(m)$ is a module for $A_n(V).$ In the case $n=0,$ 