Teoria de representacions de quivers. El teorema de Gabriel

Abstract

[en] The main goal of this work is the study of representations of quivers. A representation of a quiver is an algebraic structure that is made up of a directed graph together with vector spaces assigned to the vertices, and linear maps assigned to the arrows. First of all, basic notions of representations’ theory of quivers are given, along with some particular examples. We also define the path algebra of a quiver and some of its more elemental properties. Next we elaborate on a series of concepts and results of homological algebra (chain complexes, homologies, exact sequences, resolutions, etc.) in order to define the Tits form of a quiver. Then we present various basic definitions of algebraic geometry (affine algebraic varieties, Zariski topology, etc.) that will help us to define the representation variety of a quiver. Finally, we close this work with Gabriel’s theorem. This theorem characterizes the so-called quivers of finite representation type, i.e. those quivers which have only finitely many isomorphism classes of representations.