We investigate the cluster-tilted algebras of finite representation type over an algebraically closed field. We give an explicit description of the relations for the quivers for finite representation type. As a consequence we show that a (basic) cluster-tilted algebra of finite type is uniquely determined by its quiver. Also some necessary conditions on the shapes of quivers of cluster-tilted algebras of finite representation type are obtained along the way.
In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.
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 some extension of silted algebras are cluster-tilted algebras.
A cluster tilted algebra is known to be gentle if and only if it is cluster tilted of Dynkin type $\bbA$ or Euclidean type $\tilde{\bbA}$. We classify all finite dimensional algebras which are derived equivalent to gentle cluster tilted algebras.
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. In particular, we study which $τ$-rigid C-modules are also $τ$-rigid B-modules.
We give a complete description of all special biserial cluster-tilted algebras over a finite dimensional hereditary algebra H over an algebraically closed field K.
We give a new characterization of tilted algebras by the existence of certain special subquivers in their Auslander-Reiten quiver. This result includes the existent characterizations of this kind and yields a way to obtain more tilted quotient algebras from a given algebra.
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.
In this paper, we characterize all the finite dimensional algebras that are derived equivalent to an m-cluster tilted algebra of type A tilde. This generalizes a result of Bobonski and Buan [9].
We use the maximal faces of the $m$-cluster complex of type A to describe the m-cluster tilted algebras of type A as quivers with relations. We then classify connected components of m-cluster tilted algebras of type A up to derived equivalence using tilting complexes directly related to the combinatorics of the m-cluster complex of type A. This generalizes a result of Buan and Vatne.
In this paper, we characterize all the finite dimensional algebras that are m-cluster tilted algebras of type A tilde. We show that these algebras are gentle and we give an explicit description of their quivers with relations.
We survey recent development of the study of finite-dimensional selfinjective algebras over a field which are socle equivalent to selfinjective orbit algebras of tilted type.
We prove that if a quasi-tilted algebra is tame, then the associated moduli spaces are products of projective spaces. Together with an earlier result of Chindris this gives a geometric characterization of the tame quasi-tilted algebras.
Let $Λ$ be a hereditary algebra, $B_0=End_Λ(T_0)$ be a tilted algebra. We will construct tilting $B_0$-modules from tilting $Λ$-modules and use this result to show how tilting quivers of BB-tilted algebras can be obtained from those of $Λ$
The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.
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.
Let $Λ$ be a quasi-tilted algebra. If $Λ$ is representation-finite, it was shown by Happel, Reiten, and Smalø that $Λ$ is tilted. We provide a new, short proof of this result.
We determine the minimal lower bound $n$, with $n \geq 1$, where the $n$-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of $m$, $m \ge 2$, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the $(m+1)$-th power of …
We classify the Auslander-Reiten components of the bounded derived category of Λ, where Λ is a cluster-tilted of type Ã. The main tool is the combinatoric description of the indecomposable complexes in the bounded homotopy category via homotopy strings and homotopy bands.
We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type $\D$. They are $2$-CY-tilted algebras. Using a suitable process of mutations of quivers with potentials (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algeb…
This is an appendix to the Handbook of Tilting Theory, edited by Angeleri-Huegel, Happel and Krause, to be published soon. Part 1 of the appendix provides an outline of the core of tilting theory. Part 2 is devoted to topics where tilting modules and tilted algebras have shown to be relevant. Both Parts 1 and 2 contain historical annodations and reminiscences. The final Part 3 is a short report on some striking recent developments which are motivated by the cluster theory of Fomin and Zelevinsky. In particular, we will guide the reader to the basic properties of cluster tilted algebras, to the…
We provide an affirmative answer for the question raised almost twenty years ago concerning the characterization of tilted artin algebras by the existence of a sincere finitely generated module which is not the middle of a short chain.
In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver.Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same…
In this paper we give the derived equivalence classification of cluster-tilted algebras of type An. We show that the bounded derived category of such an algebra depends only on the number of 3-cycles in the quiver of the algebra.
The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and $T'$ are cluster-tilting objects in a cluster category $\mathcal{C}$, then $\End_{\mathcal{C}}(T)$ is isomorphic to $\End_{\mathcal{C}}(T')$ if and only if $T=τ^i T'$.
Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials. In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type $G_2$.
We prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three, and give an explicit construction of an Auslander generator of its module category. We also show that if a connected selfinjective algebra admits an acyclic generalised standard Auslander-Reiten component then its representation dimension is equal to three.
Cluster-tilted algebras are trivial extensions of tilted algebras. This correspondence induces a surjective map from tilted algebras to cluster-tilted algebras. If B is a cluster-tilted algebra, we use the fibre of B under this map to study the module category of B. In particular, we introduce the notion of reflections of tilted algebras and define an algorithm that constructs the transjective component of the Auslander-Reiten quiver of cluster-tilted algebras of tree type.
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 show how a cluster-tilted algebra of finite representation type is related to the corresponding tilted algebra, in the case of algebras defined over an algebraically closed field.
In this work we study the connection between iterated tilted algebras and m-cluster tilted algebras. We show that an iterated tilted algebra induces an m-cluster tilted algebra. This m-cluster tilted algebra can be seen as a trivial extension of another iterated tilted algebra which is derived equivalent to the original one. We give a procedure to find this new iterated tilted algebra. These m-cluster tilted algebras are quotients of higher relation extensions.
Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C, then we show that if C is constrained, or else if B is tame, then HH^1(B) is isomorphic, as a k-vector space, to the direct sum of HH^1(C) with k^{n\_{B,C}}, where n\_{B,C} is an invariant linking the bound quivers of B and C. In the representation-finite case, HH^1(B) can be read off simply by looking at the quiver of B.
In this paper, we prove that relation-extensions of quasi-tilted algebras are 2-Calabi-Yau tilted. With the objective of describing the module category of a cluster-tilted algebra of euclidean type, we define the notion of reflection so that any two local slices can be reached one from the other by a sequence of reflections and coreflections. We then give an algorithmic procedure for constructing the tubes of a cluster-tilted algebra of euclidean type. Our main result characterizes quasi-tilted algebras whose relation-extensions are cluster-tilted of euclidean type.
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.
Motivated by work of Barot, Geiss and Zelevinsky, we study a collection of Z-bases (which we call companion bases) of the integral root lattice of a root system of simply-laced Dynkin type. Each companion basis is associated with the quiver of a cluster-tilted algebra of the corresponding type. In type A, we establish that the dimension vectors of the finitely generated indecomposable modules over a cluster-tilted algebra may be obtained, up to sign, by expanding the positive roots in terms of any companion basis for the quiver of that algebra. This generalises part of Gabriel's Theorem. Also,…
We study the projective dimension of finitely generated modules over cluster-tilted algebras End(T) where T is a cluster-tilting object in a cluster category C. It is well-known that all End(T)-modules are of the form Hom(T,M) for some object M in C, and since End(T) is Gorenstein of dimension 1, the projective dimension of Hom(T,M) is either zero, one or infinity. We define in this article the ideal I_M of End(T[1]) given by all endomorphisms that factor through M, and show that the End(T)-module Hom(T,M) has infinite projective dimension precisely when I_M is non-zero. Moreover, we apply the…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a preprojective tilting module. For example, all representation-finite cluster-tilted algebras are cluster-concealed. If $C$ is a representation-finite cluster-tilted algebra, then the indecomposable $C$-modules are shown to be determined by their dimension vectors. For a general cluster-tilted algebra $C$, we are going to describe the dimension vectors of the inde…
We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq \module(A,\dd)$, the moduli space $\M(C)^{ss}_θ$ is either a point or just $\mathbb P^1$ whenever $θ$ is an integral weight for which $C^s_θ\neq \emptyse…
We provide a far reaching derived equivalence classification of the cluster-tilted algebras of Dynkin type D and suggest standard forms for the derived equivalence classes. We believe that the classification is complete, but some subtle questions remain open. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with standard forms.
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.
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 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 …
Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.
We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander-Reiten quiver of the cluster-tilted algebra, which we call local slice.
We study the projective dimensions of the restriction of functors Hom(-,X) to a contravariantly finite rigid subcategory T of a triangulated category C. We show that the projective dimension of Hom(-,X)|T is at most one if and only if there are no non-zero morphisms between objects in T[1] factoring through X, when the object X belongs to a suitable subcategory of C. As a consequence, we obtain a characterisation of the objects of infinite projective dimension in the category of finitely presented contravariant functors on a cluster-tilting subcategory of C.
In this paper, we compute the Frobenius dimension of any cluster tilted algebra of finite type. Moreover, we give conditions on the bounded quiver of a cluster tilted algebra $Λ$ such that $Λ$ has non-trivial open Frobenius structures.
We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbuesch.
We define an operation which associates to a pair (B,M) where B is a cluster-tilted algebra and M is a B-module which lies in a local slice of B, a new cluster-tilted algebra B'. In terms of the quivers, this operation corresponds to adding one vertex (and arrows).
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of…
In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras.
Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type \tilde{\mathbb{A}}.
We give an introduction to the theory of cluster categories and cluster tilted algebras. We include some background on the theory of cluster algebras, and discuss the interplay with cluster categories and cluster tilted algebras.
We prove that all gentle 2-Calabi-Yau tilted algebras (over an algebraically closed field) are Jacobian, moreover their bound quiver can be obtained via block decomposition. Related families of gentle $(m+1)$-Calabi-Yau tilted algebras are the $m$-cluster-tilted algebras of type $\mathbb{A}$ and $\widetilde{\mathbb{A}}$. For these algebras we prove that a module $M$ is stable Cohen-Macaulay if and only if $Ω^{m+1} τM \simeq M$.
We prove that indecomposable transjective modules over cluster-tilted algebras are uniquely determined by their dimension vectors. Similarly, we prove that for cluster-concealed algebras, rigid modules lifting to rigid objects in the corresponding cluster category are uniquely determined by their dimension vectors. Finally, we apply our results to a conjecture of Fomin and Zelevinsky on denominators of cluster variables.
The objective of this paper is to give a concrete interpretation of the dimension of the first Hochschild cohomology space of a cyclically oriented or tame cluster tilted algebra in terms of a numerical invariant arising from the potential.
