For any given quiver [Gamma], there is a preprojective algebra and deformed preprojective algebras associated to it. If the ground field is of characteristic 0, then there are no finite dimensional representations of deformed preprojective algebras with special weight 1. However, if the ground field is of characteristic p, there are many dimension vectors with nonempty representation spaces of that deformed preprojective algebras.

In this thesis, we study the representation category of deformed preprojective algebra with weight 1 over field of characteristic p > 0. The motivation is to count the number of rational points of the numbers X[subscript [lambda]] =[mu]⁻¹([lambda]) of moment maps at the special weights [lambda] [element of] K[superscript x] over finite fields, and to study the relations of the Zeta functions of these algebraic varieties X[subscript [lambda]] which are defined over integers to Betti numbers of the manifolds X[subscript [lambda]](C). The first step toward counting is to study the categories of representations of the deformed preprojective algebras [Pi][superscript [lambda]].

The main results of this thesis contain two types of quivers. First result shows that over finite field, the category of finite dimensional representations of deformed preprojective algebras [Pi]¹ associated to type A quiver with weight 1 is closely related to the category of finite dimensional representations of the preprojective algebra associated to two different type A quivers. Moreover, we also give the relations between their Zeta functions. The second result shows that over algebraically closed field of characteristic p > 0, the category of finite dimensional representations of deformed preprojective algebras [Pi]¹ associated to Jordan quiver with weight 1 has a close relationship with the category of finite dimensional representations of preprojective algebra associated to Jordan quiver.