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”.