Idun Reiten
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.
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Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $τ$-tilting theory: Inspired by the work of Iyama, Jørgensen and Yang \cite{ijy}, we introduce the notion of $τ$-tilting subcategories and tilting subcategories of $\mod\T$. We show that there exists …
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple mutation behaviour: Each indecomposable object has exactly d mutations. On the other hand, the weakly d-cluster tilting subcategories of C which lack functorial finiteness can have much more complicated mutation behaviour: For each 0 <= \ell <= d-1, we show a weakly d-cluster tilting subcategory T_{\ell} which has an indecomposable object with precisely \ell…
We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.
We show that every two-term tilting complex over an Artin algebra has a tilting module over a certain factor algebra as a homology group. Also, we determine the endomorphism algebra of such a homology group, which is given as a certain factor algebra of the endomorphism algebra of the two-term tilting complex. Thus, every derived equivalence between Artin algebras given by a two-term tilting complex induces a derived equivalence between the corresponding factor algebras.
We give a characterization of $n$-cluster tilting subcategories of representation-directed algebras based on the $n$-Auslander-Reiten translations. As an application we classify acyclic Nakayama algebras with homogeneous relations which admit an $n$-cluster tilting subcategory. Finally, we classify Nakayama algebras of global dimension $d<\infty$ which admit a $d$-cluster tilting subcategory.
Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U, and that, in turn, support tilting subcategories of D/U can be lifted uniquely to maximal 1-orthogonal subcategories of D.
We study higher cluster tilting objects in generalized higher cluster categories arising from dg algebras of higher Calabi-Yau dimension. Taking advantage of silting mutations of Aihara-Iyama, we obtain a class of $m$-cluster tilting objects in generalized $m$-cluster categories. For generalized $m$-cluster categories arising from strongly ($m+2$)-Calabi-Yau dg algebras, by using truncations of minimal cofibrant resolutions of simple modules, we prove that each almost complete $m$-cluster tilting $P$-object has exactly $m+1$ complements with periodicity property. This leads us to the conjectur…
We give a combinatorial classification of cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type $A_{ \infty }$.
Let T be a locally finite triangulated category with an autoequivalence F such that the orbit category T/F is triangulated. We show that if X is an m-cluster tilting subcategory, then the image of X in T/F is an m-cluster tilting subcategory if and only if X is F-perodic. We show that for path-algebras of Dynking quivers δone may study the periodic properties of n-cluster tilting objects in the n-cluster category Cn(kδ) to obtain information on periodicity of the preimage as n-cluster tilting subcategories of Db(kδ). Finally we classify the periodic properties of all 2-cluster tilting objects …
We show that $τ$-tilting finite simply connected algebras are representation-finite. Then, some related algebras are considered, including iterated tilted algebras, tubular algebras and so on. We also prove that the $τ$-tilting finiteness of non-sincere algebras can be reduced to that of sincere algebras. This motivates us to give a complete list of $τ$-tilting finite sincere simply connected 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 $Λ$
In this paper we treat the $τ$-tilting finiteness of biserial (respectively special biserial) algebras over algebraically closed (respectively arbitrary) fields. Inside these families, to compare the notions of representation-finiteness and $τ$-tilting finiteness, we reduce the problem to the $τ$-tilting finiteness of minimal representation-infinite (special) biserial algebras. Building upon the classification of minimal representation-infinite algebras, we fully determine which minimal representation-infinite (special) biserial algebras are $τ$-tilting finite and which ones are not. To do so,…
$n\mathbb{Z}$-cluster tilting subcategories are an ideal setting for higher dimensional Auslander-Reiten theory. We give a complete classification of $n\mathbb{Z}$-cluster tilting subcategories of module categories of Nakayama algebras. In particular, we show that there are three kinds of Nakayama algebras that admit $n\mathbb{Z}$-cluster tilting subcategories: finite global dimension, selfinjective and non-Iwanaga-Gorenstein. Only the selfinjective ones can admit more than one $n\mathbb{Z}$-cluster tilting subcategory. It has been shown by the second author, that each such $n\mathbb{Z}$-clust…
We present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique to recover a cluster-tilting object from its graded quiver in the cluster category of coh $\mathbb{X}$.
As the first attempt to classify $τ$-tilting finite two-point algebras, we have determined the $τ$-tilting finiteness for minimal wild two-point algebras and some tame two-point algebras.
Let $Λ$ be an artin algebra. In this paper, the notion of $n\mathbb{Z}$-Gorenstein cluster tilting subcategories will be introduced. It is shown that every $n\mathbb{Z}$-cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is $n\mathbb{Z}$-Gorenstein if and only if $Λ$ is an Iwanaga-Gorenstein algebra. Moreover, it will be shown that an $n\mathbb{Z}$-Gorenstein cluster tilting subcategory of ${\rm{mod}}{\mbox{-}}Λ$ is an $n\mathbb{Z}$-cluster tilting subcategory of the exact category ${\rm{Gprj}}{\mbox{-}}Λ$, the subcategory of all Gorenstein projective objects of ${\rm{mod}}{\mbox{-}}Λ$. So…
We give an example of a finite-dimensional algebra with a 2-cluster tilting module and a simple module which has infinite complexity. This answers a question of Erdmann and Holm.
Geometric models for endomorphism algebras of tilting modules over gentle algebras
研究温和代数上倾斜模及其自同态代数，利用标记曲面与平铺框架，证明平铺忠实剖分对应倾斜模，定义辅助代数并证其与自同态代数同构，构造新平铺，引入保持翻转的倾斜模。
We construct a representation of the blob algebra over a ring allowing base change to every interesting (i.e. non--semisimple) specialisation which, in quasihereditary specialisations, passes to a full tilting module.
In this paper we describe all indecomposable two-term partial tilting complexes over a Brauer tree algebra with multiplicity 1 using a criterion for a minimal projective presentation of a module to be a partial tilting complex. As an application we describe all two-term tilting complexes over Brauer star algebra and compute their endomorphism rings.
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
We give a characterization of radical square zero bound quiver algebras $\mathbf{k} Q/\mathcal{J}^2$ that admit $n$-cluster tilting subcategories and $n\mathbb{Z}$-cluster tilting subcategories in terms of $Q$. We also show that if $Q$ is not of cyclically oriented extended Dynkin type $\tilde{A}$, then the poset of $n$-cluster tilting subcategories of $\mathbf{k} Q/\mathcal{J}^2$ with relation given by inclusion forms a lattice isomorphic to the opposite of the lattice of divisors of an integer which depends on $Q$.
We determined the $τ$-tilting finiteness of Schur algebras over an algebraically closed field of arbitrary characteristic, except for a few small cases.
Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for the non-singular triangular or spherical algebras but not for any other weighted surface algebra with bipartite Gabriel quiver.
We give a proof, based on the rigidity of tilting complexes, that the class of self-injective finite-dimensional algebras over an algebraically closed field is closed under derived equivalence.
In this paper, we introduce the notion of $ν$-stable silting-discrete algebras, which unify silting-discrete algebras and tilting-discrete self-injective algebras, where $ν$ is a triangle auto-equivalence of the bounded homotopy category of finitely generated projective modules. Moreover, we give an example of tilting-discrete self-injective algebras which are not silting-discrete.
In this paper all two-term tilting complexes over a Brauer tree algebra with multiplicity one are described using a classification of indecomposable two-term partial tilting complexes obtained earlier in a joint paper with M. Antipov. The endomorphism rings of such complexes are computed.
Classical tilting and $τ$-tilting theory via duplicated algebras
通过复制代数将τ-倾斜理论转化为经典倾斜理论，扩展了拟单则代数的结果，并揭示了支持τ-倾斜偏序集与Bongartz区间的关系。
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…
Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.
We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each reduced expression $\bf{w}$ of $w$ and give a sufficient condition on $\bf{w}$ such that $M(\bf{w})$ is a tilting object. In particular, the stable category is triangle equivalent to the derived category of the endomorphism algebra of $M(\bf{w})$. Moreover, we compare it with a triangle equivalence given by Amiot-Reiten-Todorov for a cluster category.
Let B be a cluster-tilted algebra. We prove that B is $τ$-tilting finite if and only if B is representation-finite.
In this paper, we compute the number of two-term tilting complexes for an arbitrary symmetric algebra with radical cube zero over an algebraically closed field. Firstly, we give a complete list of symmetric algebras with radical cube zero having only finitely many isomorphism classes of two-term tilting complexes in terms of their associated graphs. Secondly, we enumerate the number of two-term tilting complexes for each case in the list.
Derived equivalent gentle algebras which are not tilting-cotilting equivalent
研究导出等价gentle代数的分类问题，构造出不满足tilting-cotilting等价的实例，丰富了gentle代数的等价分类研究
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
Let $G=SL(2,5)$ be the special linear group of $2 \times 2$-matrices with coefficients in the field with $5$ elements. We show that the principal block over a splitting field $K$ of characteristic two of the group algebra $KG$ has a $3$-cluster tilting module. This gives the first example of a representation-infinite block of a group algebra having a cluster tilting module and answers a question by Erdmann and Holm.
We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization…
We prove the existence of an $m$-cluster tilting object in a generalized $m$-cluster category which is $(m+1)$-Calabi-Yau and Hom-finite, arising from an $(m+2)$-Calabi-Yau dg algebra. This is a generalization of the result for the ${m = 1}$ case in Amiot's Ph.~D.~thesis. Our results apply in particular to higher cluster categories associated to suitable finite-dimensional algebras of finite global dimension, and higher cluster categories associated to Ginzburg dg categories coming from suitable graded quivers with superpotential.
In this paper, we investigate the relative dominant dimension with respect to an injective module and characterize the algebras with finite relative dominant dimension. As an application, we introduce the almost n-precluster tilting module and establish a correspondence between almost n-precluster tilting modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a description of the Gorenstein projective modules over almost n-minimal Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster tilting modules.
The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposabl…
We show that a gentle algebra over a field is $τ$-tilting finite if and only if it is representation-finite. The proof relies on the "brick-$τ$-tilting correspondence" of Demonet-Iyama-Jasso and on a combinatorial analysis.
We find a relationship between the global dimension of an algebra $A$ and the global dimension of the endomorphism algebra of a $τ$-tilting module, when $A$ is of finite global dimension. We show that, in general, the global dimension of the endomorphism algebra is not always finite. For monomial algebras and special biserial algebras of global dimension two, we prove that the global dimension of the endomorphism algebra of any $τ$-tilting module is always finite. Moreover, for special biserial algebras, we give an explicit bound.
Comparing the bounded derived categories of an algebra and of the endomorphism algebra of a given support τ-tilting module, we find a relation between the derived dimensions of an algebra and of the endomorphism algebra of a given τ-tilting module.
In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $τ$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.
We provide a classification of generalized tilting modules and full exceptional sequences for the dual extension algebra of the path algebra of a uniformly oriented linear quiver modulo the ideal generated by paths of length two with its opposite algebra. For the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions.
Cluster tilting modules for local algebras
该研究给出局部有限维代数中非平凡cluster tilting模的首个实例，借助GAP的QPA包通过箭图与关系显式计算对应高维Auslander代数，并探讨相关问题与猜想。
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.
A construction of support $τ$-tilting modules over $τ$-tilting finite algebras
本文研究τ-倾斜有限代数上支撑τ-倾斜模的构造，借助中间扩张函子沿recollement粘合半砖，探讨τ-倾斜有限性在模范畴粘合下的性质，并给出构造实例。
A complete classification of $τ$-tilting finite Schur algebras
本文确定经典Schur代数部分块的τ-倾斜有限性，结合相关文献结果完成τ-倾斜有限Schur代数的完全分类，还得到Schur代数S(2,r)对应块的分类及严格野生Schur代数的分类，仅余三个未解决情况。
Let $Λ$ be a finite dimensional algebra such that $\mathcal{L}_Λ$ or $\mathcal{R}_Λ\neq\emptyset$. Then $Λ$ is $τ$-tilting finite if and only if $Λ$ is representation-finite.
Let $A$ be a finite dimensional algebra over an algebraically closed field $k$. Let $T$ be a tilting $A$-module and $B={\rm End}_A\ T$ be the endomorphism algebra of $T$. In this paper, we consider the correspondence between the tilting $A$-modules and the tilting $B$-modules, and we prove that there is a one-one correspondence between the basic $T$-tilting $A$-modules in $T^{\perp}$ and the basic tilting $B$-modules in $^{\perp}(D_BT)$. Moreover, we show that there is a one-one correspondence between the $T$-contravariantly finite $T$-resolving subcategories of $T^{\perp}$ and the basic $T$-t…
We classify generalized tilting modules and full exceptional sequences for the family of quasi-hereditary quotients of type A zig-zag algebras and for a related family of algebras. We also give a characterization of these quotients as quasi-hereditary algebras with simple preserving duality that are 'close' to self-injective algebras.
We give a construction of Gorenstein projective $τ$-tilting modules in terms of tensor products of modules. As a consequence, we give a class of non-self-injective algebras admitting non-trivial Gorenstein projective $τ$-tilting modules. Moreover, we show that a finite dimensional algebra $Λ$ over an algebraically closed field is $CM$-$τ$-tilting finite if $T_n(Λ)$ is $CM$-$τ$-tilting finite which gives a partial answer to a question on $CM$-$τ$-tilting finite algebras posed by Xie and Zhang.
Let $Λ$ be an artin algebra and $\mathcal{M}$ be an n-cluster tilting subcategory of mod$Λ$. We show that $\mathcal{M}$ has an additive generator if and only if the n-almost split sequences form a basis for the relations for the Grothendieck group of $\mathcal{M}$ if and only if every effaceable functor $\mathcal{M}\rightarrow Ab$ has finite length. As a consequence we show that if mod$Λ$ has n-cluster tilting subcategory of finite type then the n-almost split sequences form a basis for the relations for the Grothendieck group of $Λ$.
In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.
We present a new way to construct $n$-cluster tilting subcategories of abelian categories. Our method takes as input a direct system of abelian categories $\mathcal{A}_i$ with certain subcategories and, under reasonable conditions, outputs an $n$-cluster tilting subcategory of an admissible target $\mathcal{A}$ of the direct system. We apply this general method to a direct system of module categories $\text{mod}Λ_i$ of representation-directed algebras $Λ_i$ and obtain an $n$-cluster tilting subcategory $\mathcal{M}$ of a module category $\text{mod}\mathcal{C}$ of a locally bounded Krull-Schmid…
The class of support $τ$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $τ$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $τ$-rigid modules. We show that $A$ is $τ$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $τ$-tilting module form a simplicial complex $Δ(A)$. We show that if $A$ is $τ…
We construct derived equivalences between group graded self-injective algebras, starting from equivalences between their 1-components, obtained via a construction of J. Rickard and S. Al-Nofayee.
$τ$-tilting finiteness of two-point algebras II
本研究提出基于$g$-向量的策略，发现新的极小$\ au$-倾斜无限两点代数，判定了含根立方零情形在内的多种两点单项式代数的$\ au$-倾斜有限性，并证明Kronecker代数的导出等价类仅含自身与反代数。
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