Cluster-tilting Objects Research Articles

The Multiplication Formulas of Weighted Quantum Cluster Functions

By applying the property of Ext-symmetry and the affine space structure of certain fibers, we introduce the notion of weighted quantum cluster functions and prove their multiplication formulas associated to abelian categories with Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects.

A CATEGORIFICATION OF ACYCLIC PRINCIPAL COEFFICIENT CLUSTER ALGEBRAS

AbstractIn earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau.

Orbifold diagrams

We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. We moreover realise this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object. This is similar to (and relies on) a result by Baur-King-Marsh for Postnikov diagrams on the disk.

On Maximal Green Sequence for Quivers Arising from Weighted Projective Lines

We investigate the existence and non-existence of maximal green sequences for quivers arising from weighted projective lines. Let Q be the Gabriel quiver of the endomorphism algebra of a basic cluster-tilting object in the cluster category \(\mathcal {C}_{\mathbb {X}}\) of a weighted projective line \(\mathbb {X}\). It is proved that there exists a quiver \(Q^{\prime }\) in the mutation equivalence class Mut(Q) of Q such that \(Q^{\prime }\) admits a maximal green sequence. Furthermore, there is a quiver in Mut(Q) which does not admit a maximal green sequence if and only if \(\mathbb {X}\) is of wild type.

Modules of infinite projective dimension

We characterize the modules of infinite projective dimension over the endomorphism algebras of Oppermann–Thomas cluster-tilting objects X in (n+2)-angulated categories (𝒞,Σn,Θ). For an indecomposable object M of 𝒞, we define in this article the ideal IM of End𝒞(ΣnX) given by all endomorphisms that factor through addM, and show that the End𝒞(X)-module Hom𝒞(X,M) has infinite projective dimension precisely when IM is nonzero. As an application, we generalize a recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.

Relative Oppermann–Thomas Cluster Tilting Objects in $$(n+2)$$-Angulated Categories

Relative Oppermann–Thomas Cluster Tilting Objects in $$(n+2)$$-Angulated Categories

Simple-minded systems and reduction for negative Calabi-Yau triangulated categories

We develop the basic properties of w w -simple-minded systems in ( − w ) (-w) -Calabi-Yau triangulated categories for w ⩾ 1 w \geqslant 1 . We show that the theory of simple-minded systems exhibits striking parallels with that of cluster-tilting objects. The main result is a reduction technique for negative Calabi-Yau triangulated categories. Our construction provides an inductive technique for constructing simple-minded systems.

Mutation of frozen Jacobian algebras

We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive categorification of cluster algebras with frozen variables via Frobenius categories. As an application, we show that the mutation of cluster-tilting objects in various such categorifications, such as the Grassmannian cluster categories of Jensen–King–Su, is compatible with Fomin–Zelevinsky mutation of quivers. We also describe an extension of this combinatorial mutation rule allowing for arrows between frozen vertices, which the quivers arising from categorifications and dimer models typically have.

The interplay between 2-Calabi-Yau and 3-Calabi-Yau triangulated categories

This paper surveys the constructions of 3-Calabi-Yau triangulated categories with simple-minded collections due to Ginzburg and Kontsevich-Soibelman and the constructionsof 2-Calabi-Yau triangulated categories with cluster-tilting objects due to Buan-Marsh-Reiten-Reineke-Todorov and Amiot, and includes a discussion on the normal form of2-Calabi-Yau triangulated categories with cluster-tilting objects.

Tilting and cluster-tilting objects on weighted projective lines

In this paper, weinvestigate tilting objects and cluster-tilting objects in the categories associated with weighted projective lines. For the case of genus one, tilting objects in the stable category of vector bundles are proved to be precisely cluster-tilting objects in the associated cluster category. Particularly, cluster-tilting objects in the associated cluster category are induced by tilting sheaves in the category of coherent sheaves.

Relative cluster tilting objects in triangulated categories

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 $\tau$-tilting $\la$-modules, which generalizes a result of Adachi-Iyama-Reiten \cite{AIR}. We develop a basic theory on $T[1]$-cluster tilting objects. In particular, we introduce a partial order on the set of $T[1]$-cluster tilting objects and mutation of $T[1]$-cluster tilting objects, which can be regarded as a generalization of `cluster-tilting mutation'. As an application, we give a partial answer to a question posed in \cite{AIR}.

Silting reduction and Calabi–Yau reduction of triangulated categories

We study two kinds of reduction processes of triangulated categories, that is, silting reduction and Calabi–Yau reduction. It is shown that the silting reduction $\mathcal {T}/\mathsf {thick}\mathcal {P}$ of a triangulated category $\mathcal {T}$ with respect to a presilting subcategory $\mathcal {P}$ can be realized as a certain subfactor category of $\mathcal {T}$, and that there is a one-to-one correspondence between the set of (pre)silting subcategories of $\mathcal {T}$ containing $\mathcal {P}$ and the set of (pre)silting subcategories of $\mathcal {T}/\mathsf {thick}\mathcal {P}$. This result is applied to show that the Amiot–Guo–Keller construction of $d$-Calabi–Yau triangulated categories with $d$-cluster-tilting objects takes silting reduction to Calabi–Yau reduction.

Graded Frobenius Cluster Categories

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 particular obtaining that degrees are additive on exact sequences. We show that the categories of Buan, Iyama, Reiten and Scott, some of which were used by Geiß, Leclerc and Schröer to categorify cells in partial flag varieties, and those of Jensen, King and Su, categorifying Grassmannians, are examples of graded Frobenius cluster categories.

Mutation of friezes

We study mutations of Conway–Coxeter friezes which are compatible with mutations of cluster-tilting objects in the associated cluster category of Dynkin type A. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. We observe how the frieze can be divided into four distinct regions, relative to the entry at which we want to mutate, where any two entries in the same region obey the same mutation rule. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton.

Internally Calabi\u2013Yau algebras and cluster-tilting objects

We describe what it means for an algebra to be internally d-Calabi–Yau with respect to an idempotent. This definition abstracts properties of endomorphism algebras of (d-1)-cluster-tilting objects in certain stably (d-1)-Calabi–Yau Frobenius categories, as observed by Keller–Reiten. We show that an internally d-Calabi–Yau algebra satisfying mild additional assumptions can be realised as the endomorphism algebra of a (d-1)-cluster-tilting object in a Frobenius category. Moreover, if the algebra satisfies a stronger ‘bimodule’ internally d-Calabi–Yau condition, this Frobenius category is stably (d-1)-Calabi–Yau. We pay special attention to frozen Jacobian algebras; in particular, we define a candidate bimodule resolution for such an algebra, and show that if this complex is indeed a resolution, then the frozen Jacobian algebra is bimodule internally 3-Calabi–Yau with respect to its frozen idempotent. These results suggest a new method for constructing Frobenius categories modelling cluster algebras with frozen variables, by first constructing a suitable candidate for the endomorphism algebra of a cluster-tilting object in such a category, analogous to Amiot’s construction in the coefficient-free case.

On cluster-tilting objects in a triangulated category with Serre duality

ABSTRACTLet 𝒟 be a Krull–Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object T. We introduce the notion of an FΛ-stable support τ-tilting module, induced by the shift functor and the Auslander–Reiten translation, in the cluster-tilted algebra . We show that there exists a bijection between basic cluster-tilting objects in 𝒟 and basic FΛ-stable support τ-tilting Λ-modules. This generalizes a result of Adachi–Iyama–Reiten [1]. As a consequence, we obtain that all cluster-tilting objects in 𝒟 have the same number of nonisomorphic indecomposable direct summands.

NEARLY MORITA EQUIVALENCES AND RIGID OBJECTS

If $T$ and $T^{\prime }$ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)); that is, their module categories are equivalent “up to a simple module”. This result has been generalized by Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly Morita equivalent, and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations.

G-stable support τ-tilting modules

Motivated by τ-tilting theory developed by T. Adachi, O. Iyama, I. Reiten, for a finite-dimensional algebra Λ with action by a finite group G; we introduce the notion of G-stable support τ-tilting modules. Then we establish bijections among G-stable support τ-tilting modules over Λ; G-stable two-term silting complexes in the homotopy category of bounded complexes of finitely generated projective Λ-modules, and G-stable functorially finite torsion classes in the category of finitely generated left Λ-modules. In the case when Λ is the endomorphism of a G-stable cluster-tilting object T over a Hom-finite 2-Calabi-Yau triangulated category C with a G-action, these are also in bijection with G-stable cluster-tilting objects in C. Moreover, we investigate the relationship between stable support τ-tilitng modules over Λ and the skew group algebra ΛG

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it mapsreachableindecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is acategorificationof the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-calledfriezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we callgeneralized friezesand that, for cluster categories of Dynkin typeA, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.