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

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