Cluster-tilted Algebras Research Articles

On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras

Assume that K is an algebraically closed field and denote by KG(R) the Krull-Gabriel dimension of R, where R is a locally bounded K-category (or a bound quiver K-algebra). Assume that C is a tilted K-algebra and Cˆ,Cˇ,C˜ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that KG(C˜)=KG(Cˇ)≤KG(Cˆ). Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that KG(C˜)=KG(Cˇ)=KG(Cˆ)∈{0,2,∞}. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.

Positive cluster complexes and τ-tilting simplicial complexes of cluster-tilted algebras of finite type

In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support -tilting modules.

Torsion pairs and cosilting in type A˜

Torsion pairs in the category of finitely presented modules over a noetherian ring can be parametrised by the class of cosilting modules. In this paper, we characterise such modules in terms of their indecomposable summands, providing a new approach to the classification of torsion pairs. In particular, we classify cosilting modules over cluster-tilted algebras of type A ˜ . We do this by using a geometric model for finite- and infinite-dimensional modules over such algebras.

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.

Open Frobenius Cluster-Tilted Algebras

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra [Formula: see text] such that [Formula: see text] has non-trivial open Frobenius structures.

Zavadskij modules over cluster-tilted algebras of type $ \mathbb{A} $

<abstract><p>Zavadskij modules are uniserial tame modules. They arose from interactions between the poset representation theory and the classification of general orders. The main problem is to characterize Zavadskij modules over a finite-dimensional algebra $ A $. In this setting, we prove that the indecomposable uniserial $ A $-modules with a mast of multiplicity one in each vertex are Zavadskij modules. Since the Zavadskij property carries over to direct summands, but it is not invariant under the formation of direct sums, we give a criterion to determine when the direct sum of indecomposable Zavadskij modules is again a Zavadskij module. In addition, we use the triangulations of the $ n+3 $-gon associated with the cluster-tilted algebra of type $ \mathbb{A}_{n} $ to give a formula for the number of indecomposable Zavadskij modules over any cluster-tilted algebra of type $ \mathbb{A}_{n} $. In this case, the formula gives the dimension of the cluster-tilted algebra. As an application, we discuss some integer sequences in the OEIS (The On-Line Encyclopedia of Integer Sequences) that allow us to enumerate indecomposable Zavadskij modules.</p></abstract>

Universal Deformation Rings of Modules over Self-injective Cluster-Tilted Algebras Are Trivial

Let [Formula: see text] be a fixed algebraically closed field of arbitrary characteristic, let [Formula: see text] be a finite dimensional self-injective [Formula: see text]-algebra, and let [Formula: see text] be an indecomposable non-projective left [Formula: see text]-module with finite dimension over [Formula: see text]. We prove that if [Formula: see text] is the Auslander–Reiten translation of [Formula: see text], then the versal deformation rings [Formula: see text] and [Formula: see text] (in the sense of F.M. Bleher and the second author) are isomorphic. We use this to prove that if [Formula: see text] is further a cluster-tilted [Formula: see text]-algebra, then [Formula: see text] is universal and isomorphic to [Formula: see text].

τ-Tilting finite cluster-tilted algebras

We prove if B is a cluster-tilted algebra, then B is τB-tilting finite if and only if B is representation-finite.

Extending silted algebras to cluster-tilted algebras

It is well known that the relation extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove that some extension of silted algebras are cluster-tilted algebras.

On indecomposable τ-rigid modules for cluster-tilted algebras of tame type

For a given cluster-tilted algebra A of tame type, it is proved that different indecomposable τ-rigid A-modules have different dimension vectors. This is motivated by Fomin and Zelevinsky's denominator conjecture for cluster algebras. As an application, we establish a weak version of the denominator conjecture for cluster algebras of tame type. Namely, we show that different cluster variables have different denominators with respect to a given cluster for a cluster algebra of tame type. Our approach involves Iyama and Yoshino's construction of subfactors of triangulated categories. In particular, we obtain a description of the subfactors of cluster categories of tame type with respect to an indecomposable rigid object, which is of independent interest.

Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite type

Let $\Lambda$ be a cluster-tilted algebra of finite type over an algebraically closed field and $B$ be one of the associated tilted algebras. We show that the $B$-modules, ordered form right to left in the Auslander-Reiten quiver of $\Lambda$ form a maximal forward hom-orthogonal sequence of $\Lambda$-modules whose dimension vectors form the $c$-vectors of a maximal green sequence for $\Lambda$. Thus we give a proof of Igusa-Todorov's conjecture.

τ-Rigid modules from tilted to cluster-tilted algebras

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra where E is the C-C-bimodule . In particular, we study which τC-rigid C-modules are also τB-rigid B-modules.

Maximal green sequences for cluster-tilted algebras of finite representation type

We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin–Zelevinsky mutation of the extended exchange matrix, (2) a complete forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author’s work with P. J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.

Hall algebras as Poisson algebras

Motivated by Joyces work on motivic Hall algebras, we observe that for an algebra $\Lambda$ with Hall polynomials,the Hall algebra $\mathcal{H}(\Lambda)$ of $\Lambda$ admits a natural structure of Poisson algebra. For a given antisymmetric bilinear form over its Grothendieck group $\go(\Lambda)$satisfying certain condition, there is a homomorphism of Poisson algebras from the Hall algebra $\mathcal{H}(\Lambda)$ to its associated torus algebra. In particular, the condition holds true forrepresentation-finite cluster-tilted algebras.

Gorenstein Properties of Simple Gluing Algebras

For given bound quiver algebras A and B, we obtain a new algebra Λ, called the simple gluing algebra, by identifying two vertices. We investigate the Gorenstein homological property, the singularity category, the Gorenstein defect category and the Cohen-Macaulay Auslander algebra of Λ in terms of that of A and B. Finally, we give applications to cluster-tilted algebras.

Strongness of companion bases for cluster-tilted algebras of finite type

For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.

Derived equivalences and stable equivalences of Morita type, II

We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broué’s abelian defect group conjecture. Another motivation is a conjecture by Auslander and Reiten on stable equivalences preserving the number of non-projective simple modules. A machinery is presented to construct lifts for a large class of algebras, including Frobenius-finite algebras introduced in this paper. In particular, every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. Consequently, the Auslander–Reiten conjecture is true for stable equivalences of Morita type between Frobenius-finite algebras. Examples of Frobenius-finite algebras are abundant, including representation-finite algebras, Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we show that, for a Nakayama-stable idempotent element e in an algebra A over an algebraically closed field, each tilting complex over eAe can be extended to a tilting complex over A that induces an almost ν-stable derived equivalence studied in the first paper of this series. Moreover, the machinery is applicable to verify Broué’s abelian defect group conjecture for several cases mentioned in the literature.

Projective dimensions and extensions of modules from tilted to cluster-tilted algebras

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B=C⋉E where E is the C-C-bimodule ExtC2(DC,C). We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete characterization of the projective dimension of a C-module inside modB. If a C-module M satisfies ExtC1(M,M)=0, we show two sufficient conditions for M to satisfy ExtB1(M,M)=0. In particular, if MC is indecomposable and ExtC1(M,M)=0, we prove MB always satisfies ExtB1(M,M)=0.

Lattice Properties of Oriented Exchange Graphs and Torsion Classes

The exchange graph of a 2-acyclic quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. When the quiver admits a nondegenerate Jacobi-finite potential, the exchange graph admits a natural acyclic orientation called the oriented exchange graph, as shown by Br\"ustle and Yang. The oriented exchange graph is isomorphic to the Hasse diagram of the poset of functorially finite torsion classes of a certain finite dimensional algebra. We prove that lattices of torsion classes are semidistributive lattices, and we use this result to conclude that oriented exchange graphs with finitely many elements are semidistributive lattices. Furthermore, if the quiver is mutation-equivalent to a type A Dynkin quiver or is an oriented cycle, then the oriented exchange graph is a lattice quotient of a lattice of biclosed subcategories of modules over the cluster-tilted algebra, generalizing Reading's Cambrian lattices in type A. We also apply our results to address a conjecture of Br\"ustle, Dupont, and P\'erotin on the lengths of maximal green sequences.

Modules over cluster-tilted algebras that do not lie on local slices

We characterize the indecomposable transjective modules over an arbitrary cluster-tilted algebra that do not lie on a local slice, and we provide a sharp upper bound for the number of (isoclasses of) these modules.