The cluster algebras were introduced by Fomin-Zelevinsky in a paper which appeared 10 years ago. There are connections to many different areas of mathematics, including quiver representations. One direction of research has been to try to model the ingredients in the definition of cluster algebras in “nice” categories, like module categories or related categories.
In this lecture we illustrate the idea and use of “categorification” by concentrating on only one ingredient in the definition of cluster algebras. This is the operation of quiver mutation, which we define. For finite quivers (i.e. directed graphs) without oriented cycles this leads to the so-called cluster
categories, which are modifications of certain module categories. For other
types of quivers some stable categories of maximal Cohen-Macaulay modules over commutative Gorenstein rings are the relevant categories.
We start the lecture with background material on quiver representations.