This seminar provides an introduction to quiver representation theory, in its algebraic and geometric aspects.
In the first part, we introduce the basic theory of quiver representations. We will construct the path algebra associated with a given quiver, whose modules correspond to the representations of the quiver itself. The discussion will include explicit examples of quivers and path algebras along with a classification of their representations.
In the second part, we turn to quiver varieties, defined as moduli spaces that encode the theory of quiver representations. We will characterize points in these varieties both geometrically (as closed orbits in a vector space) and algebraically (as semisimple representations). Finally, we will present the Le Bruyn-Procesi theorem, which provides a method for constructing generators of the coordinate ring of a quiver variety.