Introduction to Quivers

Abstract

Quivers are directed graphs which are commonly used in fields such as representation theory and noncommutative geometry. This paper is meant to provide a short introduction for quivers and algebras produced from those quivers, called path algebras. We first look at basic definitions of quivers Q and path algebras kQ. We also cover some algebraic properties of path algebras in order to have a better understanding of the category of finite representations of a quiver Q. In fact, such category is equivalent the category of finitely generated left kQ-module corresponding to the quiver Q. As an example, we briefly describe how to obtain a representation of Q from a left kQ-module. At the end, we take a look at a bounded quiver Q (a.k.a. a quiver Q with a set of relations R) and its path algebra kQ / I where I is a two sided ideal generated by R. We use the Beilinson quiver for $\mathscr {P}^2$ with relations as an example to illustrate the bounded quiver and its corresponding path algebra.