http://www.maths.leeds.ac.uk/~marsh/research_articles/pp18.pdf

Tilting theory and cluster combinatorics

We establish basic properties of cluster algebras associated with oriented bordered surfaces with marked points. In particular, we show that the underlying cluster complex of such a cluster algebra does not depend on the choice of coefficients, describe this complex explicitly in terms of "tagged triangulations" of the surface, and determine its homotopy type and its growth rate.

/pdf/cluster-algebras-and-triangulated-surfaces-part-i-cluster-3jmotnp470.pdf

Cluster algebras and triangulated surfaces. Part I: Cluster complexes

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

/pdf/on-triangulated-orbit-categories-4k4m3u3a5b.pdf

On triangulated orbit categories.

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

/pdf/cluster-tilted-algebras-4zqgolxuuk.pdf

Cluster-tilted algebras

We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

/pdf/cluster-structures-for-2-calabi-yau-categories-and-unipotent-z8fzhy7zac.pdf

Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.

/pdf/cluster-structures-for-2-calabi-yau-categories-and-unipotent-1rk5pa7bv5.pdf

Aslak Bakke Buan

Papers

Tilting theory and cluster combinatorics

Tilting theory and cluster combinatorics

Cluster-tilted algebras

Cluster structures for 2-Calabi-Yau categories and unipotent groups

Cluster structures for 2-Calabi-Yau categories and unipotent groups