From iterated tilted algebras to cluster-tilted algebras
Michael Barot, Elsa Fernández, María Inés Platzeck, Nilda Isabel Pratti, Sonia Trepode
本文研究了迭代倾斜代数和簇倾斜代数与关系扩张之间的关系。在Dynkin的情况下，它表明，关系是非常强大的组合。
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.
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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.
Enumerating iterated tilted algebras in type $A$
该研究建立了A型$A_n$迭代倾斜代数同构类与凸$n+1$边形旋转等价非交叉生成树的双射关系，通过构造迭代倾斜代数与例外集的关联完成证明。
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.
We compute the Hochschild cohomology groups of the cluster-tilted algebras of finite representation type.
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 classify all finite dimensional algebras which are derived equivalent to m-cluster tilted algebras of type A.
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.
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.
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 $Λ$
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].
Cluster-tilted algebras of finite representation type
研究了代数闭域上有限表示型的簇倾斜代数。我们给出了有限表示型的箭图的关系的明确描述。作为结果，我们证明了一个有限型的（基本）簇-倾斜代数是由它的群唯一确定的。同时也得到了有限表示型簇倾斜代数箭图形状的一些必要条件。
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.
We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.
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.
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.
Derived equivalence classification of m-cluster tilted algebras of type A
我们利用A型$m$ -簇复形的极大面把A型m-簇倾斜代数描述为具有关系的箭图。然后，我们分类的连通组件的m-集群倾斜代数A型导出等价使用倾斜复合物直接相关的组合的m-集群复杂的A型。这推广了Buan和Vatne的一个结果。
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…
Some Remarks Concerning Tilting Modules and Tilted Algebras. Origin. Relevance. Future. (An appendix to the Handbook of Tilting Theory)
这是即将出版的《倾斜理论手册》的附录，由安杰莱里-休格尔、哈佩尔和克劳斯编辑。附录的第一部分提供了倾斜理论核心的概要。第2部分是专门的主题，倾斜模块和倾斜代数已被证明是相关的。第1和第2部分都包含历史注释和回忆。最后的第3部分是一个简短的报告，对一些引人注目的最近的发展是由集群理论的福明和Zelevinsky的动机。特别是，我们将引导读者的基本性质集群倾斜代数，倾斜代数和集群倾斜代数之间的关系，而且集群类别提供了一个普遍的设置，所有相关的倾斜和集群倾斜代数。此外，我们将把注意力集中到集群倾斜对象的复杂性，并表现出这种复杂性的初步描述。
Let B be a cluster-tilted algebra. We prove that B is $τ$-tilting finite if and only if B is representation-finite.
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…
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 …
Derived equivalence classification for cluster-tilted algebras of type $A_n$
本文给出An型簇倾斜代数的导出等价分类。证明了这类代数的有界导出范畴只依赖于代数的环中3-圈的个数。
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.
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.
Counting cluster-tilted algebras of type $A_n$
本文通过计算具有基图$A_n$的任意群的变异类，给出了$A_n$型非同构簇倾斜代数的个数的一个显式公式.还将得出，如果$T$和$T'$是簇类别$\mathcal{C}$中的簇倾斜对象，则$\End_{\mathcal{C}}(T)$同构于$\End_{\mathcal{C}}(T')$当且仅当$T=τ^i T'$。
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.
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 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 consider $m$-cluster tilted algebras arising from quivers of Euclidean type and we give necessary and sufficient conditions for those algebras to be representation finite. For the case $\widetilde{A}$, using the geometric realization, we get a description of representation finite type in terms of $(m+2)$-angulations. We establish which $m$-cluster tilted algebras arise at the same time from quivers of type $A$ and $\widetilde{A}$. Finally, we characterize representation infinite $m$-cluster tilted algebras arising from a quiver of type $\widetilde{A}$ as $m$-relations extensions of some ite…
From tilted to cluster-tilted algebras of Dynkin type
在定义于代数闭域上的代数中，我们证明了有限表示型的簇倾斜代数与相应的倾斜代数之间的关系。
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 …
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.
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.
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $τ^2 T\cong T$, where $τ$ is the Auslander-Reiten translation in $D^b(\operatorname{coh}\mathbb{X})$. As an application of this result, we classify the 2-representation-finite algebras which are derived-equivalent to a canonical algebra. This complements Iyama-Oppermann's classification of the iterated tilted 2-representation-finite algebras. By passing to 3-preprojective algebras, we…
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.
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 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…
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.
Derived equivalences of self-injective 2-Calabi--Yau tilted algebras
本文证明2-Calabi--Yau稳定范畴中具有自入射自同态代数的极大刚性对象的自同态代数导出等价，并给出诱导该等价的双边倾斜复形描述。
Cluster-tilted algebras and slices
我们给出了一个标准，允许验证是否两个倾斜代数有相同的关系扩展（从而对应于相同的集群倾斜代数）。这个标准是在一个组合配置的Auslander-Reiten的集群倾斜代数，我们称之为本地切片。
We show the existence of Hall polynomials for representation-finite cluster-tilted algebras.
The Self-injective Cluster Tilted Algebras
我们将确定所有的自内射簇倾斜代数。它们都是有限表示型和特殊的二进制数。
On the Galois coverings of a cluster-tilted algebra
研究了簇倾斜代数的某个Galois覆盖的模范畴，称之为簇重复代数。我们的主要结果比较了倾斜代数C的簇重复代数和C的重复代数在Hughes和Waschbuesch意义下的模范畴。
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 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 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 apply the notion of a full convex subcategory to a wide range of algebras including tilted, quasi-tilted, shod, weakly shod, left and right glued, laura, simply connected, strongly simply connected, left supported, and cluster-tilted. In particular, given an algebra $Λ$ from one of the aforementioned classes, we investigate certain factor algebras $Λ/I$ where $I$ is an ideal generated by a suitable idempotent.
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $τ$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $τ$-tilting theory. Afterwards we define $τ$-slices and prove that complete slices of tilted algebras and local slices of cluster tilted algebras are examples of complete $τ$-slices. Then we apply this concept to the study of simply connected tilted algebras. Finally, we study the one-point extensions and the split-by-nilpotent extensions of an algebra with $τ$-slices.
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 $Ω^{m+1} τM \simeq M$.
Let $Λ$ be a 1-Auslander-Gorenstein Algebra. We give a necessary and sufficient condition for $Λ$ to be a tilted algebra.
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).
We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as suppleme…
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