Ralf Schiffler
We give a geometric realization of cluster categories of type $D_n$ using a polygon with $n$ vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
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In this note, we consider the $d$-cluster-tilted algebras, the endomorphism algebras of $d$-cluster-tilting objects in $d$-cluster categories. We show that a tilting module over such an algebra lifts to a $d$-cluster-tilting object in this $d$-cluster category.
Y. Palu has generalized the cluster multiplication formulas to 2-Calabi-Yau categories with cluster tilting objects (\cite{Palu2}). The aim of this note is to construct a variant of Y. Palu's formula and deduce a new version of the cluster multiplication formula (\cite{XiaoXu}) for acyclic quivers in the context of cluster categories.
We prove the existence of cluster characters for Hom-infinite cluster categories. For this purpose, we introduce a suitable mutation-invariant subcategory of the cluster category. We sketch how to apply our results in order to categorify any skew-symmetric cluster algebra. More applications and a comparison to Derksen-Weyman-Zelevinsky's results will be given in a future paper.
We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi-Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi-Yau triangulated category. This generalizes a recent work of D. Smith for cluster categories.
Group actions on relative cluster categories and Higgs categories
本文研究有限群作用下的相对簇范畴和Higgs范畴，通过轨道突变和聚类特征链接到具有系数的非单向簇代数，提供非单向簇代数的加法分类法。
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a hereditary abelian category $\mathcal{H}$, we verify that the tilting functor Hom$_\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\mathcal{C(H)}$ to the cluster category $\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\mathc…
Dyck paths categories are introduced as a combinatorial model of the category of representations of quivers of Dynkin type An. In particular, it is proved that there is a bijection between some Dyck paths and perfect matchings of some snake graphs. The approach allows us to give formulas for cluster variables in cluster algebras of Dynkin type An in terms of Dyck paths.
Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not al…
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 the 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.
We construct the intermediate coverings of cluster-tilted algebras by defining the generalized cluster categories. These generalized cluster categories are Calabi-Yau triangulated categories with fraction CY-dimension and have also cluster tilting objects (subcategories). Furthermore we study the representations of these intermediate coverings of cluster-tilted algebras.
A correspondence between additive and monoidal categorifications with application to Grassmannian cluster categories
本文提出簇代数的加法与幺半范畴化的猜想对应，通过格拉斯曼簇范畴构造A型通用核，并应用该对应在格拉斯曼簇范畴中构造刚性与非刚性不可分解模。
In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi-Yau property of the cluster category.
We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated rep…
We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the $m$th power of a translation quiver and show how it can be used to realise the $m$-cluster category in terms of the cluster category.
We show that the m-cluster category of type D_n 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.
Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories are actually just quotients of cluster categories. The other half can be obtained as quotients of 2-cluster categories, the "lowest" type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.
The goals of this expository article are on one hand to describe how to construct ($m$-) cluster categories from triangulations (resp. from $m+2$-angulations) of polygons. On the other hand, we explain how to use translation quivers and their powers to obtain the $m$-cluster categories directly from the diagonals of a polygon.
Let $A$ be a hereditary algebra. We construct a fundamental domain for the cluster category of $A$ inside the category of modules over the duplicated algebra $\bar{A}$ of $A$. We then prove that there exists a bijection between the tilting objects in the cluster category and the tilting $\bar{A}$-modules all of whose non projective-injective indecomposable summands lie in the left part of the module category of $\bar{A}$.
Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposa…
Let A be a hereditary algebra over an algebraically closed field. We prove that an exact fundamental domain for the m-cluster category of A is the m-left part of the m-replicated algebra $A^{(m)}$ of A. Moreover, we obtain a one-to-one correspondence between the tilting objects in the m-cluster category (that is, the m-clusters) and those tilting $A^{(m)}$-modules for which all non projective-injective direct summands lie in the m-left part of $A^{(m)}$.
In this paper, we show that the repetitive cluster category of type $D_n$, defined as the orbit category $\mathcal{D}^b(\mathrm{mod}K D_n)/(τ^{-1}[1])^p$, is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, \cite{S}, which we recover when $p=1$.
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
Cluster characters for 2-Calabi-Yau Frobenius extriangulated categories
本文定义带丛倾斜对象的 2-Calabi–Yau Frobenius 外三角范畴的丛特征标，统一了三角与正合两类范畴中的既有理论。
We show that the quotient of the continuous cluster category $\mathcal C_π$ modulo the additive subcategory generated by any cluster is an abelian category and we show that it is isomorphic to the category of infinite length modules over the endomorphism ring of the cluster. These theorems extend the theorems of Caldero-Chapoton-Schiffler and Buan-Marsh-Reiten for cluster categories to the continuous cluster category of type $A$. These results will be generalized in a series of forthcoming joint papers of the two authors with Job Rock.
We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.
In arXiv:1209.0038 we constructed topological triangulated categories C_c as stable categories of certain topological Frobenius categories F_c. In this paper we show that these categories have a cluster structure for certain values of c including c=pi. The continuous cluster categories are those C_c which have cluster structure. We study the basic structure of these cluster categories and we show that C_c is isomorphic to an orbit category D_r/F_s of the continuous derived category D_r if c=r pi/s. In C_pi, a cluster is equivalent to a discrete lamination of the hyperbolic plane. We give the r…
We prove the existence of an $m$-cluster tilting object in a generalized $m$-cluster category which is $(m+1)$-Calabi-Yau and Hom-finite, arising from an $(m+2)$-Calabi-Yau dg algebra. This is a generalization of the result for the ${m = 1}$ case in Amiot's Ph.~D.~thesis. Our results apply in particular to higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension, and higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential.
We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of $m$-cluster tilting objects in generalized $m$-cluster categories. For generalized $m$-cluster categories arising from strongly ($m+2$)-Calabi-Yau dg algebras, by using truncations of minimal cofibrant resolutions of simple modules, we prove that each almost complete $m$-cluster tilting $P$-object has exactly $m+1$ complements with periodicity property. This leads us to the conjectur…
We give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type $A_{ \infty }$.
Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory.
We introduce a class of categories, called \emph{clustered hyperbolic categories}, which are generated by equivalent categories of representations of some Weyl cluster algebras. Every preseed $p$ gives rise to a \emph{categorical preseed} $\mathcal{P}$ which generates a clustered hyperbolic category through \emph{categorical mutations}. A "categorification" of Weyl cluster algebra is introduced in the sense of defining a map $\mathbf{F}_{p}$ from the clustered hyperbolic category $\mathfrak{C}(\mathcal{P})$ to the Weyl cluster algebra $\mathcal{H}(p)$ where image of $\mathbf{F}_{p}$ generates …
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.
We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class A_n, D_n, E_6, E_7 or E_8 are triangulated equivalent to u-cluster categories of the corresponding Dynkin type. The proof relies on the 'Morita' theorem for u-cluster categories by Keller and Reiten, along with the recent computation of Calabi-Yau dimensions of stable module categories by Dugas.
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the bounded derived category $\mathcal{D}^{b}(H)$, and of the cluster-tilting objects, in terms of the category $\modΓ$\ of finitely generated modules over a suitable tilted algebra $% Γ.$ Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.
We study the maximal rigid subcategories in $2-$CY triangulated categories and their endomorphism algebras. Cluster tilting subcategories are obviously maximal rigid; we prove that the converse is true if the $2-$CY triangulated categories admit a cluster tilting subcategory. As a generalization of a result of [KR], we prove that any maximal rigid subcategory is Gorenstein with Gorenstein dimension at most 1. Similar as cluster tilting subcategory, one can mutate maximal rigid subcategories at any indecomposable object. If two maximal rigid objects are reachable via mutations, then their endom…
We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.
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 …
In this paper, we use the stable categories of some selfinjective algebras to describe the singularity categories of the cluster-tilted algebras of Dynkin type. Furthermore, in this way, we settle the problem of singularity equivalence classification of the cluster-tilted algebra of type $A$, $D$ and $E$ respectively.
We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This generalized cluster structure is different from those introduced by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras from surfaces, we describe the generalized cluster structure of the orbit space in terms of a triangulated orbifold. In this case, we give a complete list of exchange polynomials, and we classify the algebras of rank 1 …
Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra with vanishing differentials and study its cluster category. We show that this DG algebra is sign-twisted Calabi-Yau, and realize its cluster category as a triangulated hull of an orbit category of a derived category, and as the singularity category of a finite dimensional Iwanaga-Gorenstein algebra. Along the way, we give two results which stand on their own. First, we show that the derived category of coherent sheaves over a Calabi-Yau algebra has a natural cluster tilting subcategory whose dimension is determined by …
Associated with some finite dimensional algebras of global dimension at most 2, a generalized cluster category was introduced in \cite{Ami3}, which was shown to be triangulated and 2-Calabi-Yau when it is $\Hom$-finite. By definition, the cluster categories of \cite{Bua} 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 \cite{Bua2}, are triangle equivalent to generalized cluster categories. This was already shown for some special elements in \cite{Ami3}.
The Caldero-Chapoton formula relates for hereditary algebras of Dynkin type the cluster characters of the end terms of an Auslander-Reiten sequence with the cluster character of the middle term. We extend this result to generalized cluster categories with cluster tilting object by considering Auslander-Reiten triangles.
Recently the first author studied multi-gradings for generalised cluster categories, these being 2-Calabi-Yau triangulated categories with a choice of cluster-tilting object. The grading on the category corresponds to a grading on the cluster algebra without coefficients categorified by the cluster category and hence knowledge of one of these structures can help us study the other. In this work, we extend the above to certain Frobenius categories that categorify cluster algebras with coefficients. We interpret the grading K-theoretically and prove similar results to the triangulated case, in p…
We put cluster tilting in ageneral framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an abelian structure. These abelian quotients turn out to be module categories of Goreisten algebras of dimension at most one.
We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.
Tropical coefficient dynamics for higher-dimensional cluster categories
本文研究高维簇范畴中指标的突变规律，证明其遵循高维版本的热带系数动力学。
In this survey we discuss some combinatorial aspects of higher cluster categories.
We consider triangulated orbit categories, with the motivating example of cluster categories, in their usual context of algebraic triangulated categories, then present them from another perspective in the framework of topological triangulated categories.
In this paper we give a geometric-combinatorial description of the cluster categories of type E. In particular, we give an explicit geometric description of all cluster tilting objects in the cluster category of type E_6. The model we propose here arises from combining two polygons, and it generalises the description of the cluster category of type A and D.
We study the cluster combinatorics of $d-$cluster tilting objects in $d-$cluster categories. By using mutations of maximal rigid objects in $d-$cluster categories which are defined similarly 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 $d+1$ triangles of $d-$cluster tilting objects in [IY], we prove that any almost complete $d-$cluster tilting object has exactly $d+1$ complements, compute the extension groups between these complements, and study the middle terms of these $d+1…
Relative cluster categories and Higgs categories
本文将簇范畴构造推广至相对语境，证明了由左(n+1)-Calabi--Yau态射产生的Higgs范畴中存在稳定n-Calabi--Yau的n-cluster tilting对象，并应用于相对Ginzburg dg代数与高维Auslander代数。
Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are a generalization of cluster-tilting objects. When $\D$ is $2$-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let $\la={\rm End}^{op}_{\D}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\D$ and support $τ$-tilting $\la$-modules, which gener…
We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius categories we consider are the categories of finitely generated Gorenstein projective modules over the singular Nakajima category associated to a Dynkin diagram and their standard Frobenius quotients. In particular, we are able to categorify all finite-type skew-symmetric cluster algebras with universal coefficients and finite-type Grassmannian cluster algebras. Along the way, we classify the standard Frobenius models of a certain famil…
On geometric models in representation theory
本文探讨了几何模型在表示论中的应用，特别是温和代数的表面模型对导出范畴和Fukaya范畴的贡献。
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and injectives has a cluster tilting subcategory M, then B/M is abelian. More precisely, it is equivalent to the category of finitely presented modules over the stable category of M.
In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty}$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a completion of these discrete cluster categories in the sense of Paquette-Yildirim.
This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting subcategory with neither loops nor 2-cycles do have the generalized version of cluster structure. This is in particular the case of cluster categories over non-algebraically closed fields.
Negative Calabi-Yau discrete cluster categories via Nakayama representations and persistence theory
本文利用持续性理论构造对称 Nakayama 表示的无限离散版本，经稳定化得到一族负 Calabi-Yau 三角范畴，并刻画其几何模型与 AR 理论。
Cyclic poset are generalizations of cyclically ordered sets. In this paper we show that any cyclic poset gives rise to a Frobenius category over any discrete valuation ring R. The continuous cluster categories of arXiv:1209.1879 are examples of this construction. If we twist the construction using an admissible automorphism of the cyclic poset, we generate other examples such as the m-cluster category of type A-infinity (m>2).
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