Tilting Theory and Cluster Algebras

The purpose of this chapter is to give an introduction to the theory of cluster categories and cluster-tilted algebras, with some background on the theory of cluster algebras, which motivated these topics. We will also discuss some of the interplay between cluster algebras on one side and cluster categories/cluster-tilted algebras on the other, as well as feedback from the latter theory to cluster algebras. The theory of cluster algebras was initiated by Fomin and Zelevinsky [FZ1], and further developed by them in a series of papers, including [FZ2], some involving other coauthors. This theory has in recent years had a large impact on the representation theory of algebras. The first connection with quiver representations was given in [MRZ]. Then the cluster categories were introduced in [BMRRT] in order to model some of the ingredients in the definition of a cluster algebra. For this purpose a tilting theory was developed in the cluster category. (See [CCS1] for the independent construction of a category in the An case which turned out to be equivalent to the cluster category [CCS2]). This led to the theory of cluster-tilted algebras initiated in [BMR1] and further developed in many papers by various authors. The theory of cluster-tilted algebras (and cluster categories) is closely connected with ordinary tilting theory. Much of the inspiration comes from usual tilting theory. Features missing in tilting theory when trying to model clusters from the theory of cluster algebras made it necessary to replace the category modH of finitely generated H-modules for a finite dimensional hereditary algebra H with a related category which is now known as the cluster category. On the other hand, the theory of cluster-tilted algebras provides a new point of view on the old tilting theory. The Bernstein-Gelfand-Ponomarev (BGP) reflection functors were an important source of inspiration for the development of tilting theory, which provided a major generalization of the work in [BGP]. The Fomin-Zelevinsky (FZ) mutation, which is an essential ingredient in the definition of cluster algebras, gives a generalization of these reflections in another direction. We start with introducing cluster algebras in the first section. We illustrate the essential concepts with an example, which will be used throughout the chapter. We give main results and conjectures about cluster algebras which are relevant for our further discussion. In Section 2 we introduce and investigate cluster categories, followed by cluster-tilted algebras in Section 3 . In Section 4 we discuss the interplay between cluster algebras and cluster categories/cluster-tilted algebras, and we also give applications to cluster algebras. The cluster categories are a special case of the more general class of Hom-finite triangulated Calabi-Yau categories of dimension 2 (2-CY categories), and much of the theory generalizes to this setting. Denote by modΛ the category of finitely generated (left) modules over a finite dimensional