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 :
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