Representations of deformed preprojective algebras and quantum groups

Abstract

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.