2.8. Quivers 19
(a) What is the center of A for different q? If q is not a root of
unity, what are the two-sided ideals in A?
(b) For which q does this algebra have finite dimensional repre-
sentations?
Hint: Use determinants.
(c) Find all finite dimensional irreducible representations of A for
such q.
Hint: This is similar to part (c) of the previous problem.
2.8. Quivers
Definition 2.8.1. A quiver Q is a directed graph, possibly with
self-loops and/or multiple edges between two vertices.
Example 2.8.2.


We denote the set of vertices of the quiver Q as I and the set
of edges as E. For an edge h E, let h , h denote the source and
target of h, respectively:

h
h

h
Definition 2.8.3. A representation of a quiver Q is an assign-
ment to each vertex i I of a vector space Vi and to each edge h E
of a linear map xh : Vh −→ Vh .
It turns out that the theory of representations of quivers is a part
of the theory of representations of algebras in the sense that for each
quiver Q, there exists a certain algebra PQ, called the path algebra
of Q, such that a representation of the quiver Q is “the same” as
a representation of the algebra PQ. We shall first define the path
algebra of a quiver and then justify our claim that representations of
these two objects are “the same”.
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