In this chapter, we show that to each finite dimensional algebra over an algebraically closed field K corresponds a graphical structure, called a quiver, and that, conversely, to each quiver corresponds an associative K-algebra, which has an identity and is finite dimensional under some conditions. Similarly, as will be seen in the next chapter, using the quiver associated to an algebra A, it will be possible to visualise a (finitely generated) A-module as a family of (finite dimensional) K-vector spaces connected by linear maps (see Examples (I.2.4)–(I.2.6)). The idea of such a graphical representation seems to go back to the late forties (see Gabriel, Grothendieck, and Thrall) but it became widespread in the early seventies, mainly due to Gabriel. In an explicit form, the notions of quiver and linear representation of quiver were introduced by Gabriel in. It was the starting point of the modern representation theory of associative algebras.

Quivers and path algebras

This first section is devoted to defining the graphical structures we are interested in and introducing the related terminology. We shall then be able to show how one can associate an algebra to each such graphical structure and study its properties.

1.1. Definition. A quiverQ = (Q0, Q1, s, t) is a quadruple consisting of two sets: Q0 (whose elements are called points, or vertices) and Q1 (whose elements are called arrows), and two maps s, t : Q1 → Q0 which associate to each arrow α ∈ Q1 its sources(α) ∈ Q0 and its targett(α) ∈ Q0, respectively.