$\C{\S}\sharp \C[G]$ except that $\C\S$ in our setting is a right $\C[G]$-module instead of left $\C[G]$-module in the setting of crossed product. So one could have used the results from \[S\], \[BCM\] and \[DT\] to conclude that $A_{\alpha}(G,\S)$ is an associative algebra after a careful identifying two different settings. Since the proof is not very long we chose to give a complete proof.]{}

[*If we take ${\S}=\{e(g)|g\in G\}$ to be the dual basis of $\C[G]^*$ which is a right $G$-set via $e(g)\cdot h=e(h^{-1}gh)$ for $g,h\in G$ we get the associative algebra $D_{\alpha}(G)$ in \[DM5\] which is a deformation of the Drinfeld’s double $D(G).$ It turns out that there exists $\alpha$ determined naturally by the “twisted representations” such that $D_{\alpha}(G)$ is the right algebra in the study of holomorphic orbifold conformal field theory (see \[DVVV\] and \[DM5\]).*]{}

For each $s\in {\S}$, let $G_s=\{h\in G|s\cdot h=s\}$ be the stabilizer of $s.$ Note that $Res_{G_s}^{G}\a_s$ is in $Z^2(G_s,\C^*).$ Let ${\O}_s$ be the orbit of $s$ under $G$ and $G=\bigcup_{i=1}^kG_sg_i$ be a right coset decomposition with $g_1=1.$ Then ${\O}_s=\{sg_i|i=1,...,k\}$ and $G_{s\cdot g_i}=g_i^{-1}G_sg_i.$ We define several subspaces of ${\A}_{\a}(G,{\S}):$ $$S(s)=\<a\o e(s)|a\in G_{s}\>\ \ ,\ \ N(s)=\<a\o e(s)|a\in G\setminus G_{s}\>,$$ $$D(s)=\<a\o e(s)|a\in G\>\ \ ,\ \ 
