$A_0(V)$ reduces to the associative algebra $A(V)$ introduced previously in \[Z\]. The main results concerning $A_n(V)$ are summarized in Section 4. The result that we often use in this paper is the following: $M$ is irreducible if and only if each $M(n)$ is a simple $A_n(V)$-module for all $n\geq 0$ \[DM4\]. This key result allows us to reduce an infinite dimensional problem ($M$ is infinite dimensional) to a finite dimensional problem (each $M(n)$ is finite dimensional in our main theorems).

This paper is organized as follows. In Section 2 we review twisted group algebras. Section 3 is about the algebra $A_{\alpha}(G,\S)$ based on a finite group $G,$ a right $G$-set $\S$ and a suitable 2-cocycle $\alpha$ on $G.$ We construct all simple modules for $A_{\alpha}(G,\S)$ explicitly by using twisted group algebras. A basis for the center is also given for the future study. In Section 4 we recall the various notions of modules for a vertex operator algebra and review the theory on associative algebras $A_n(V).$ Section 5 is the first part on duality. We show that if $M$ is an irreducible $V$-module which is $G$-stable in the sense that $M\circ g\cong M$ for all $g,$ then there exists a 