A quiver is an oriented graph which might contain multiple edges and loops. Sometimes it is assumed to be finite.

The terminology is used in representation-theory of finite dimensional algebras, where one considers functors from this graph, viewed as a category, to the category of vector spaces. To a quiver one can associate its path algebra, which is the $k$-algebra with basis given by all paths in the quiver and multiplication given by concatenation (or zero). These path algebras are basic hereditary algebras. And Gabriel's theorem says that up to Morita equivalence all finite dimensional algebras over an algebraically closed field arise as quotients of these path algebras.

Important books to learn more about the subject include:

  • Assem, Simson, Skowronski: Elements of the representation theory of associative algebras
  • Auslander, Reiten, Smalø: Representation theory of Artin algebras
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