Representation theory of algebras and Categorification

Starting from the combinatorics of cluster algebras of finite type it was soon realized that acyclic cluster algebras are intimately related to the representation theory of hereditary algebras of the corresponding type [87], and that clusters should correspond (roughly) to tilting modules. This was formalized by the discovery of cluster categories as the orbit category of the derived category of a hereditary algebra by Buan, Marsh, Reineke, Reiten and Todorov [22], and Keller [76]. In a series of papers [23,25,31], ... culminating in [26] it was then shown that for each quiver $Q$ without oriented cycles there is a so called cluster character from the cluster category $\mathcal{C}_Q$ to the corresponding acyclic cluster algebra $\mathcal{A}_Q$, producing for example natural bijections between clusters and basic cluster tilting objects. A main tool in this approach is the natural interplay between cluster categories and cluster tilted algebras [25], [78] which also shed a new light on classical tilting theory [99] and which motivated new questions (and answers) in representation theory: Assem, Brüstle, Schiffler, (Todorov): [4,2,3,5]; Buan, Reiten et al.: [24,28,27,30]; Holm and Jørgensen: [73,72]; Ringel: [98]; Zhu: [113,114,117,115,116].

On the other hand, Geiss, Leclerc and Schröer managed to categorify the cluster algebra structure on the coordinate ring of a maximal unipotent subgroup $N$ of a complex simply connected simple Lie group via the module category of the preprojective algebra of the corresponding type, [66], based on previous work [65,61,64]. This approach uses heavily Lusztig's construction of the semicanonical basis [86] and yields similar correspondences as above in the acyclic case. It can be modified in order to categorify also the cluster structure on multihomogenous coordinate rings of partial flag varieties [62], or coordinate rings of certain unipotent cells for Kac-Moody groups [21,63].

For such ``categorification projects'' an adequate language was found in [21]. It is based on (stably resp. triangulated) 2-Calabi-Yau categories with a cluster tilting object. These categories have been introduced by Keller and Reiten [78], [77] and were further studied by students of Keller: Amiot [1], Tabuada [105], Palu [90], and by Iyama and Reiten [74].

One should note that in this new branch of representation theory most of the classical methods for representation theory of finite dimensional algebras have found new, highly non-trivial applications. This includes Auslander-Reiten theory (and their recent generalizations by O. Iyama), tilting theory, coverings, representations of quivers, weighted projective lines, triangulated categories and last but not least dg-categories. On the other hand this approach has provided new interesting cluster structures and helped to settle the conjecture on the linear independence of cluster monomials in many cases.