Derived categories and preprojective algebras

The aim of this Master Thesis is to study the structure of the preprojective algebras through the use of the derived categories. The work deals initially with the study of algebras and starts with the definition of the representation of a quiver or, equivalently, of an algebra. Since, in general, the study of algebras turns out to be very difficult, the idea behind my thesis is to consider algebras as modules over themselves and thus study the property of the category of modules over the algebra itself. In particular, thanks to the Krull-Schmidt theorem, we can reduce our study to that of indecomposable modules and thanks to Gabriel theorem we only consider quivers of Dynkin type because they are the ones whose associated module category has a finite number of indecomposable modules up to isomorphism. Moreover, thanks to Morita theory, we can reduce our field of research to the finitely dimensional module category over a basic algebra. It turns out that there always exists a quiver such that the quotient of the associated path algebra by an admissible ideal represents the basic algebra. The most important theory that give an algorithm that allows us to find indecomposable modules and to study the morphisms between them is the Auslander-Reiten Theory. Through this theory I will emphasize the constant presence of the derived Ext functor, which is one of the main reasons why, in order to deepen and generalize the study of algebras, we introduce the structure of derived categories. Thus, in the second chapter, we define them and we describe the structures we can find inside as, for example, the triangulated and the Frobenius ones. We can note that these structures expand the concepts introduced in the Auslander-Reiten theory and, in particular, thanks to a result by Happel we are able to give a good description of the derived category of the module category. Once we have laid all the groundwork regarding the theory of representations of algebras,in the third chapter we introduce preprojective algebras giving three different definitions, based on algebraic structures that are also completely different. One of the most important results in the thesis will be the equivalence between these three definitions. After stating some properties about preprojective algebras, as for example the self-injectivity and the 2-Calabi-Yau structure of the stable module category we try to generalize these concepts by introducing the higher preprojective algebras. They, surprisingly, will have properties analogous to those stated for the base case