2 Continuous Cluster Categories I

Inventiones mathematicae, 2008

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra A of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for A. We prove new results and formulate conjectures on 'good basis' properties, positivity, denominator theorems and toric degenerations.

Arxiv preprint math/0610512, 2006

We show that the m-cluster category of type Dn is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm − m + 1-gon. This generalises a result of Schiffler for m = 1. We use the notion of the mth power of a translation quiver to realise the m-cluster category in terms of the cluster category.

Algebra Colloquium, 2018

We introduce a class of categories, called clustered hyperbolic categories, which are constructed from a categorized version of preseeds called categorical preseeds using categorical mutations that are a “functorial” edition of preseed mutations. Every Weyl preseed p gives rise to a categorical preseed [Formula: see text] which generates a clustered hyperbolic category; this is formed by copies of categories each one of which is equivalent to the category of representations of the Weyl cluster algebra [Formula: see text]. A “categorical realization” of Weyl cluster algebra is provided in the sense of defining a map Fp from any clustered hyperbolic category induced from p to the Weyl cluster algebra [Formula: see text], where the image of Fp generates [Formula: see text].

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 A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy 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, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.

2016

Associated with a finite dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in [Ami08]. It was shown to be triangulated, and 2-Calabi-Yau when it is Hom-finite. By definition, the cluster categories of [BMR + 06] are a special case. In this paper we show that a large class of 2-Calabi-Yau triangulated categories, including those associated with elements in Coxeter groups from [BIRS09a], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in [Ami08] and then more generally for c-sortable elements in [AIRT10].

São Paulo Journal of Mathematical Sciences

In this paper, we show that the repetitive cluster category of type Dn, defined as the orbit category D b (modkDn)/(τ −1 [1]) p , is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, [23], which we recover when p = 1.

Advances in Mathematics, 2011

Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in [Ami08], which was shown to be triangulated and 2-Calabi-Yau when it is Hom-finite. By definition, the cluster categories of [BMR + 06] are a special case. In this paper we show that a large class of 2-Calabi-Yau triangulated categories, including those associated with elements in Coxeter groups from [BIRS09a], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in [Ami08].

Journal of Algebra, 2007

We show that a certain orbit category considered by Keller encodes the combinatorics of the m-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the combinatorics of the clusters of Fomin and Zelevinsky. This allows us to give typeuniform proofs of certain results of Fomin and Reading in the simply laced cases.

Milan Journal of Mathematics, 2008

The structure of cluster categories [BMRRT] is well suited for the combinatorics of cluster algebras [FZ1] with the main correspondence being between tilting objects and clusters. Furthermore it was shown in [IOTW] that there is a close relation between domains of virtual semi-invariants and simplicial complexes associated to cluster categories. Also the same simplicial complexes associated to cluster categories are related to the Igusa-Orr pictures in the homology of nilpotent groups.

Journal of Algebra - J ALGEBRA, 2009

We study the cluster combinatorics of d-cluster tilting objects in d-cluster categories. Using mutations of maximal rigid objects in d-cluster categories, which are defined in a similar way to mutations for d-cluster tilting objects, we prove the equivalences between d -cluster tilting objects, maximal rigid objects and complete rigid objects. Using the chain of d+1d+1 triangles of d-cluster tilting objects in [O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen–Macaulay modules, Invent. Math. 172 (1) (2008) 117–168], we prove that any almost complete d -cluster tilting object has exactly d+1d+1 complements, compute the extension groups between these complements, and study the middle terms of these d+1d+1 triangles. All results are the extensions of corresponding results on cluster tilting objects in cluster categories established for d-cluster categories in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. 204 (2006) 572–618]. They are applied to the Fomin–Reading generalized cluster complexes of finite root systems defined and studied in [S. Fomin, N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005) 2709–2757; H. Thomas, Defining an m-cluster category, J. Algebra 318 (2007) 37–46; K. Baur, R. Marsh, A geometric description of m-cluster categories, Trans. Amer. Math. Soc. 360 (2008) 5789–5803; K. Baur, R. Marsh, A geometric description of the m -cluster categories of type DnDn, preprint, arXiv:math.RT/0610512; see also Int. Math. Res. Not. 2007 (2007), doi:10.1093/imrn/rnm011], and to that of infinite root systems [B. Zhu, Generalized cluster complexes via quiver representations, J. Algebraic Combin. 27 (2008) 25–54].