Soliton Solutions on Noncommutative Orbifold $T^{2N}/G$

In this paper, we construct the common eigenstates of "translation" operators $\{U_{s}\}$ and establish the generalized $Kq$ representation on integral noncommutative torus $T^{2N}$. We then study the finite rotation group $G$ in noncommutative space as a mapping in the $Kq$ representation and prove a Blocking Theorem. We finally obtain the complete set of projection operators on the integral noncommutative orbifold $T^{2N}/G$ in terms of the generalized $Kq$ representation. Since projectors are soliton solutions on noncommutative space in the limit $\alpha ^{\prime}B_{ij}\to \infty (\Theta_{ij}/\alpha ^{\prime}\to 0)$, we thus obtain all soliton solutions on that orbifold $T^{2N}/G$.