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 ? 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 Пλ(Г, I). It is shown that those representations given by extending indecomposable representations of (Г, I) are all simple representations of Пλ(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.

The transition matrix between PBW basis and semicanonical basis of U+(sl_n(C))

We prove that the transition matrix between a special Poincaré-Birkhoff-Witt(PBW)basis and the semicanonical basis of U+(sln(C))is upper triangular and unipotent under any order which is compatible with the partial order deg.