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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