Generalized Cluster Category Research Articles

Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras

In this article, we prove that for a finite quiver Q the equivalence class of a potential up to formal change of variables of the complete path algebra \(\widehat{{{\mathbb {C}}}Q}\), is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is a noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.

Quivers with potentials for Grassmannian cluster algebras

Abstract We consider a quiver with potential (QP) $(Q(D),W(D))$ and an iced quiver with potential (IQP) $(\overline {Q}(D), F(D), \overline {W}(D))$ associated with a Postnikov Diagram D and prove that their mutations are compatible with the geometric exchanges of D. This ensures that we may define a QP $(Q,W)$ and an IQP $(\overline {Q},F,\overline {W})$ for a Grassmannian cluster algebra up to mutation equivalence. It shows that $(Q,W)$ is always rigid (thus nondegenerate) and Jacobi-finite. Moreover, in fact, we show that it is the unique nondegenerate (thus rigid) QP by using a general result of Geiß, Labardini-Fragoso, and Schröer (2016, Advances in Mathematics 290, 364–452). Then we show that, within the mutation class of the QP for a Grassmannian cluster algebra, the quivers determine the potentials up to right equivalence. As an application, we verify that the auto-equivalence group of the generalized cluster category ${\mathcal {C}}_{(Q, W)}$ is isomorphic to the cluster automorphism group of the associated Grassmannian cluster algebra ${{\mathcal {A}}_Q}$ with trivial coefficients.

Negative cluster categories from simple minded collection quadruples

Fomin and Zelevinsky’s definition of cluster algebras laid the foundation for cluster theory. The various categorifications and generalizations of the original definition led to Iyama and Yoshino’s generalized cluster categories coming from positive-Calabi-Yau triples Jin later defined simple minded collection quadruples where the special case is the analogue of Iyama and Yang’s triples: negative-Calabi-Yau triples. In this paper, we further study the quotient categories coming from simple minded collection quadruples. Our main result uses limits and colimits to describe Hom-spaces over in relation to the easier to understand Hom-spaces over Moreover, we apply our theorem to give a different proof of a result by Jin: if we have a negative-Calabi-Yau triple, then is a negative cluster category.

The cluster category of a surface with punctures via group actions

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient by this action recovers the original surface. We show that the group acts on the quivers with potentials associated to the surfaces, and that their Ginzburg dg algebras are skew group algebras of each other, up to Morita equivalence. We then use these results to construct functors between the generalized cluster categories associated to the triangulations. This allows us to give a complete description of the indecomposable objects of these categories in terms of curves on the surface, when the surface has punctures and non-empty boundary.

On cluster categories of weighted projective lines with at most three weights

Let X be a weighted projective line and CX the associated cluster category. It is known that CX can be realized as a generalized cluster category of quiver with potential. In this note, under the assumption that X has at most three weights or is of tubular type, we prove that if the generalized cluster category C(Q,W) of a Jacobi-finite non-degenerate quiver with potential (Q,W) shares a 2-CY tilted algebra with CX, then C(Q,W) is triangle equivalent to CX. As a byproduct, a 2-CY tilted algebra of CX is determined by its quiver provided that X has at most three weights. In the end, for any weighted projective line X with at most three weights, we also obtain a realization of CX via Buan-Iyama-Reiten-Scott's construction of 2-CY categories arising from preprojective algebras.

Decorated marked surfaces (part B): topological realizations

We study categories associated to a decorated marked surface $${\mathbf {S}}_\bigtriangleup $$ , which is obtained from an unpunctured marked surface $$\mathbf {S}$$ by adding a set of decorating points. For any triangulation $$\mathbf {T}$$ of $${\mathbf {S}}_\bigtriangleup $$ , let $$\Gamma _\mathbf {T}$$ be the associated Ginzburg dg algebra. We show that there is a bijection between reachable open arcs in $${\mathbf {S}}_\bigtriangleup $$ and the reachable rigid indecomposables in the perfect derived category $${\text {per}}\,\Gamma _\mathbf {T}$$ . This is the dual of the bijection, between simple closed arcs in $${\mathbf {S}}_\bigtriangleup $$ and reachable spherical objects in the 3-Calabi-Yau category $${\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$$ , constructed in the prequel (Qiu in Math Ann 365:595–633, 2016). Moreover, we show that Amiot’s quotient $${\text {per}}\,\Gamma _\mathbf {T}/{\mathcal {D}}_{fd}(\Gamma _\mathbf {T})$$ that defines the generalized cluster categories corresponds to the forgetful map $${\mathbf {S}}_\bigtriangleup \rightarrow \mathbf {S}$$ (forgetting the decorating points) in a suitable sense.

Jacobian algebras with periodic module category and exponential growth

Recently it was proven by Geiss, Labardini-Fragoso and Schröer in [1] that every Jacobian algebra associated to a triangulation of a closed surface S with a collection of marked points M is tame and Ladkani proved in [2] these algebras are (weakly) symmetric. In this work we show that for these algebras the Auslander–Reiten translation acts 2-periodically on objects. Moreover, we show that excluding only the case of a sphere with 4 (or less) punctures, these algebras are of exponential growth. These results imply that the existing characterization of symmetric tame algebras whose non-projective indecomposable modules are Ω-periodic, has at least a missing class (see [3, Theorem 6.2] or [4]).As a consequence of the 2-periodical actions of the Auslander–Reiten translation on objects, we have that the Auslander–Reiten quiver of the generalized cluster category C(S,M) consists only of stable tubes of rank 1 or 2.

Tagged mapping class groups: Auslander–Reiten translation

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface $\mathbf{S}$ with marked points and non-empty boundary, which generalizes Br\"{u}stle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category $\mathcal{D}(\Gamma_{\mathbf{S}})$ associated to the surface and the corresponding Seidel-Thomas braid group of $\mathcal{D}(\Gamma_{\mathbf{S}})$ is empty, unless $\mathbf{S}$ is a polygon with at most one puncture (i.e. of type A or D).

Cluster equivalence and graded derived equivalence.

In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are particularly interested in the question how much information about an algebra is preserved in its generalized cluster category, or, in other words, how closely two algebras are related if they have equivalent generalized cluster categories. Our approach makes use of the cluster-tilting objects in the generalized cluster categories: We first observe that cluster-tilting objects in generalized cluster categories are in natural bijection with cluster-tilting subcategories of derived categories, and then prove a recognition theorem for the latter. Using this recognition theorem we give a precise criterion when two cluster equivalent algebras are derived equivalent. For a given algebra we further describe all the derived equivalent algebras which have the same canonical cluster tilting object in their generalized cluster category. Finally we show that in general, if two algebras are cluster equivalent, then (under certain conditions) the algebras can be graded in such a way that the categories of graded modules are derived equivalent. To this end we introduce mutation of graded quivers with potential, and show that this notion reflects mutation in derived categories.

A Caldero–Chapoton formula for generalized cluster categories

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.

Algebras of acyclic cluster type: Tree type and type Ã

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

Algebras of acyclic cluster type: Tree type and typeÃ

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

Coloured quivers for rigid objects and partial triangulations: the unpunctured case

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi–Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalized cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and we give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi–Yau case. We further show that Iyama–Yoshino reduction can be interpreted as cutting along an arc in the surface.

A derived equivalence between cluster equivalent algebras

Let Q be an acyclic quiver. Associated with any element w of the Coxeter group of Q, triangulated categories Sub ̲ Λ w were introduced in Buan et al. (2009) [BIRS09]. For any reduced expression w of w, the categories Sub ̲ Λ w are shown to be triangle equivalent to generalized cluster categories C Γ w associated to algebras Γ w of global dimension ⩽2 in Amiot et al. (2011) [ART11]. For w satisfying a certain property, called co- c-sortable, other algebras A w of global dimension ⩽2 are constructed in Amiot (2009) [Ami09] and Amiot et al. (2011) [AIRT11] with a triangle equivalence C A w ≃ Sub ̲ Λ w . The main result of this paper is that the algebras Γ w and A w are derived equivalent when w is co- c-sortable. The proof constructs explicitly a tilting module using the 2-APR-tilting theory introduced in Iyama and Oppermann (2011) [IO09].

Algebras from surfaces without punctures

We introduce a new class of finite-dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invariant of Avella-Alaminos and Geiss for surface algebras and we provide a geometric model for the module category of surface algebras.

Cluster characters II: a multiplication formula

Let 𝒞 be a Hom-finite triangulated 2-Calabi–Yau category with a cluster-tilting object. Under some constructibility assumptions on 𝒞 which are satisfied, for instance, by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, we prove a multiplication formula for the cluster character associated with any cluster-tilting object. This formula generalizes those obtained by Caldero–Keller for representation finite path algebras and by Xiao–Xu for finite-dimensional path algebras. We prove an analogous formula for the cluster character defined by Fu–Keller in the set-up of Frobenius categories. It is similar to a formula obtained by Geiss–Leclerc–Schröer in the context of preprojective algebras.

The ubiquity of generalized cluster categories

Associated with a finite-dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2-Calabi–Yau when it is Hom -finite. By definition, the cluster categories of Buan et al. (2006) [4] 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 Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].