Representations of deformed preprojective algebras and quantum groups

Let (Г, I) be the bound quiver of a cyclic quiver whose vertices correspond to the Abelian group ℤ _d . In this paper, we list all indecomposable representations of (θ, I) and give the conditions that those representations of them can be extended to representations of deformed preprojective algebra П^λ(Г, I). It is shown that those representations given by extending indecomposable representations of (Г, I) are all simple representations of П^λ(Г, I). Therefore, it is concluded that all simple representations of restricted quantum group Ū _q (sl ₂) are realized in terms of deformed preprojective algebra. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10671016, 10771014), the Beijing Natural Science Foundation (Grant No. 1062003) and Science and Technology Program of Beijing Education Committee (Grant No. KM200710005013)