2.1. What is representation theory? 7

A quiver is an oriented graph Q (which we will assume to be

finite). A representation of Q over a field k is an assignment of

a k-vector space Vi to every vertex i of Q and of a linear operator

Ah : Vi → Vj to every directed edge h going from i to j (loops and

multiple edges are allowed). We will show that a representation of a

quiver Q is the same thing as a representation of a certain algebra

PQ called the path algebra of Q. Thus one may ask: what are the

indecomposable finite dimensional representations of Q?

More specifically, let us say that Q is of finite type if it has

finitely many indecomposable representations.

We will prove the following striking theorem, proved by P. Gabriel

in early 1970s:

Theorem 2.1.2. The finite type property of Q does not depend on

the orientation of edges. The connected graphs that yield quivers of

finite type are given by the following list:

• An :

• Dn:

• E6

• E7

• E8