Derived dimensions of representation-finite algebras

Derived equivalences for $Φ$-Auslander-Yoneda algebras

In this paper, we introduce $Φ$-Auslander-Yoneda algebras in a triangulated category with $Φ$ a parameter set in $\mathbb N$, and provide a method to construct new derived equivalences between these $Φ$-Auslander-Yoneda algebras (not necessarily Artin algebras), or their quotient algebras, from a given almost $ν$-stable derived equivalence. As consequences of our method, we have: (1) Suppose that $A$ and $B$ are representation-finite, self-injective Artin algebras with $_AX$ and $_BY$ additive generators for $A$ and $B$, respectively. If $A$ and $B$ are derived-equivalent, then the $Φ$-Ausland…

某些结合代数的导出范畴的不可分解性

Indecomposables of the derived categories of certain associative algebras

本文描述了结合代数的一个分支类的导出范畴的不可分解对象。这类属于这样已知的代数类温柔代数，斜温柔代数和某些退化的管状代数。

Derived equivalences between triangular matrix algebras

In this paper we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras, and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.

Jordan Hölder theorems for derived module categories of piecewise hereditary algebras

A Jordan Hölder theorem is established for derived module categories of piecewise hereditary algebras. The resulting composition series of derived categories are shown to be independent of the choice of bounded or unbounded derived module categories, and also of the choice of finitely generated or arbitrary modules.

Filiform Lie algebras with low derived length

We construct, for any integer n greater than or equal to 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n. We also give examples of n-dimensional filiform Lie algebras with derived length greater than 3.

Examples of finite dimensional algebras which do not satisfy the derived Jordan--Hölder property

We construct a matrix algebra $Λ(A,B)$ from two given finite dimensional elementary algebras $A$ and $B$ and give some sufficient conditions on $A$ and $B$ under which the derived Jordan--Hölder property (DJHP) fails for $Λ(A,B)$. This provides finite dimensional algebras of finite global dimension which do not satisfy DJHP.

导出等价下的自内射代数

Self-injective algebras under derived equivalences

本文研究导出等价对自内射 Artin 代数的影响，证明其 Nakayama 置换共轭，并以初等方法表明有限维代数的弱对称性和自内射性在导出等价下保持。

Nakayama代数导出范畴上的Braid作用

Braid action on derived category of Nakayama algebras

本文构造了一个辫群在一个对称Nakayama代数的导出范畴上的作用，该代数也是一个没有例外顶点的Brauer星代数。我们将此作用与Schaps和Zakay-Illouz定义的Brauer星代数上的仿射辫群作用联系起来。我们证明了对于没有例外顶点的Brauer星代数，作用是忠实的。

Artin代数的导出维数与表示距离

The derived dimensions and representation distances of artin algebras

该研究推广Igusa-Todorov代数提出(m,n)-Igusa-Todorov代数概念，给出其构造方法，建立一般Artin代数的导出维数与表示距离的关系，且该结果可优化部分代数的导出维数上界。

镜像反射代数的导出等价

Derived equivalences for mirror-reflective algebras

本文证明镜像反射代数的构造保持 gendo-对称代数的导出等价，即若两个 gendo-对称代数导出等价，则其对应的镜像反射代数也导出等价。

曲dg代数的bar导出范畴

The bar derived category of a curved dg algebra

曲A-无限代数在自然界中表现为dg代数的变形。我们发展的基本理论，弯曲的A-无限代数，特别是弯曲的dg代数。我们调查他们的联系与一个合适的类的dg余代数通过酒吧建设和生产奎伦模型结构上的模块类别。定义了曲dg代数的相对导出范畴的类比。

Ortho-symmetric modules, Gorenstein algebras and derived equivalences

A new homological symmetry condition is exhibited that extends and unifies several recently defined and widely used concepts. Applications include general constructions of tilting modules and derived equivalences, and characterisations of Gorenstein properties of endomorphism rings.

中心化子矩阵代数的导出等价与稳定等价

Derived and stable equivalences of centralizer matrix algebras

该研究通过矩阵初等因子定义方阵新等价关系，将主中心化子矩阵代数的各类范畴等价归约为线性代数问题，证明了置换矩阵对应代数的Morita等价与导出等价等价等结论。

On algebras of strongly derived unbounded type

Let $A$ be a finite-dimensional algebra over an algebraically closed field. We prove $A$ is a strongly derived unbounded algebra if and only if there exists an integer $m$, such that $C_m(\proj A)$, the category of all minimal projective complexes with degree concentrated in $[0, m]$, is of strongly unbounded type, which is also equivalent to the statement the repetitive algebra $\hat{A}$ is of strongly unbounded representation type. As a corollary, we can establish the dichotomy on the representation type of $C_m(\proj A)$, the homotopy category $K^b(\proj A)$ and the repetitive algebra $\hat…

Derived equivalences, restriction to self-injective subalgebras and invariance of homological dimensions

Derived equivalences between finite dimensional algebras do, in general, not pass to centraliser (or other) subalgebras, nor do they preserve homological invariants of the algebras, such as global or dominant dimension. We show that, however, they do so for large classes of algebras described in this article. Algebras $A$ of $ν$-dominant dimension at least one have unique largest non-trivial self-injective centraliser subalgebras $H_A$. A derived restriction theorem is proved: A derived equivalence between $A$ and $B$ implies a derived equivalence between $H_A$ and $H_B$. Two methods are devel…

Indecomposables with smaller cohomological length in the derived category of gentle algebras

Bongartz and Ringel proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly study this "no gaps" theorem for the bounded derived module category $D^b(A)$ of a gentle algebra $A$: if there is an indecomposable object in $D^b(A)$ of cohomological length $l>1$, then there exists an indecomposable with cohomological length $l-1$.

Derived categories of skew-gentle algebras and orbifolds

Skew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of orbifold surfaces establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their sin…

Cohomology rings, differential graded algebras and derived equivalences

In this paper, we will consider derived equivalences for differential graded endomorphism algebras by Keller's approaches. First we construct derived equivalences of differential graded algebras which are endomorphism algebras of the objects from a triangle in the homotopy category of differential graded algebras. We also obtain derived equivalences of differential graded endomorphism algebras from a standard derived equivalence of finite dimensional algebras. Moreover, under some conditions, the cohomology rings of these differential graded endomorphism algebras are also derived equivalent. T…

Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras

Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.

导tame代数与导wild代数

Derived tame and derived wild algebras

证明了代数闭域上的有限维代数是导自驯服的或导自野性的。我们还证明了导tame代数的任何变形都是导tame的。

Derived Picard groups of selfinjective Nakayama algebras

In our proceeding paper a generating set of the derived Picard group of a selfinjective Nakayama algebra was constructed combining some previous results for Brauer tree algebras and the technique of orbit categories developed there. In this paper we finish the computation of the derived Picard group of a selfinjective Nakayama algebra.

On the existence of a derived equivalence between a Koszul algebra and its Yoneda algebra

In this paper we focus on the relations between the derived categories of a Koszul algebra and its Yoneda algebra, in particular we want to consider the cases where these categories are triangularly equivalent. We prove that the simply connected Koszul algebras are derived equivalent to their Yoneda algebras. We consider derived discrete Koszul algebras, and we give necessary and sufficient conditions for these Koszul algebras to be derived equivalent to their Yoneda algebras. Finally, we look at the Koszul algebras such that they are derived equivalent to a hereditary algebra. In the case tha…

Derived Hall algebras of one-cycle gentle algebras: The infinite global dimension case

We describe, in terms of generators and relations, the derived Hall algebras associated to the one-cycle gentle algebras of infinite global dimension.

Derived tame quadratic string algebras

In this paper we determine the derived representation type of quadratic string algebras and we prove that every derived tame quadratic string algebra whose quiver has cycles is derived equivalent to some skewed-gentle algebra.

导出离散代数的导出等价是标准的

Derived equivalences for the derived discrete algebras are standard

本文证明了导出离散代数之间的导出等价等同于通过双侧投射复形的导出张量积。

导出自同构的导出离散代数的熵与多项式熵

Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras

本文研究导出离散代数的导出自同构的熵和多项式熵，定义并计算了这两种熵的值。

导皮卡德群是局部代数群

The Derived Picard Group is a Locally Algebraic Group

设A是代数闭域K上的有限维代数.导出的Picard群DPic（A）是A模同构上的双侧倾斜复形群。证明了DPic（A）是一个局部代数群，其单位连通区为Out^0（A）.若B是导Morita等价代数，则DPic（A）与DPic（B）同构为局部代数群.我们的结果扩展，我们的基础上，工作的Huisgen-Zimmermann，Saorin和Rouquier。

The algebras derived equivalent to gentle 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.

Derived Equivalence in Surface Algebras of Genus 0 via Graded Equivalence

We determine some of the derived equivalences of a class of gentle algebras called surface algebras. These algebras are constructed from an unpunctured Riemann surface of genus 0 with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. In particular, we fix a triangulation of a surface and determine when different cuts produce derived equivalent algebras.

Brauer-Thrall type theorems for derived module categories

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and strongly derived unbounded algebras naturally. The first and second Brauer-Thrall type theorems for the bounded derived category of a finite-dimensional algebra over an algebraically closed field are obtained. The first Brauer-Thrall type theorem says that derived bounded algebras are just derived finite algebras. The second Brauer-Thrall type theorem says that an …

Derived tame Nakayama algebras

We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.

Hopf代数由其幺半导出范畴决定

Hopf algebras are determined by their monoidal derived categories

该研究证明有限维Hopf代数的规范等价与有界导出范畴的幺半三角等价等价，并推广到局部有限张量Grothendieck范畴的幺半导出等价可诱导幺半阿贝尔等价。

导Tame局部代数与两点代数

Derived tame local and two-point algebras

本文确定了代数闭域上完全非线性生成的局部代数和两点代数的导出表示类型。

Derived equivalence classification of symmetric algebras of polynomial growth

We complete the derived equivalence classification of all symmetric algebras of polynomial growth, by solving the subtle problem of distinguishing the standard and nonstandard nondomestic symmetric algebras of polynomial growth up to derived equivalence.

半导出 Ringel-Hall 代数与奇周期相对导出范畴的 Hall 代数

Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

本文针对遗传阿贝尔范畴定义半导出 Ringel-Hall 代数，给出其自然基与生成关系刻画，并证明奇周期情形下相对导出范畴的导出 Hall 代数可嵌入其中。

Hochschild Cohomology and the Derived Class of M-Cluster Tilted Algebras of Type A

We classify all finite dimensional algebras which are derived equivalent to m-cluster tilted algebras of type A.

平方为radical的代数的导出范畴

Derived categories for algebras with radical square zero

确定了根式平方为零的代数的导表示类型，并给出了它们的有界导出范畴中不可分解对象的描述。

Jordan-Hölder theorems for derived categories of derived discrete algebras

For any positive integer $n$, $n$-derived-simple derived discrete algebras are classified up to derived equivalence. Furthermore, the Jordan-Hölder theorems for all kinds of derived categories of derived discrete algebras are obtained.

Derived categories of graded gentle one-cycle algebras

Let $A$ be a graded algebra. It is shown that the derived category of dg modules over $A$ (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded $A$-modules. This is applied to study derived categories of graded gentle one-cycle algebras.

The derived equivalence classification of gentle two-cycle algebras

We complete the derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski-Malicki.

对称代数的导出范畴的等价性

Equivalences of derived categories for symmetric algebras

我们给出了有限群代数（或其它对称代数）的块的导出范畴的对象集的一个特征，这些对象集在导出范畴的等价下作为单模的象集出现.我们给出了证明某些块具有等价的导出范畴的一些应用。

Brauer 图代数的导出等价

Derived equivalences of Brauer graph algebras

本文综述 Brauer 图代数的导出等价分类及其在导出等价下封闭性的证明脉络。

国内型非标准自内射代数的导等价分类

Derived equivalence classification of nonstandard selfinjective algebras of domestic type

本文给出了代数闭域上所有非标准表示-无限国内自内射代数的一个完全导出等价分类。作为结果，也得到了这些代数的完全稳定等价分类。

Derived class of m-cluster tilted algebras of type A tilde

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].

Rigidity of tilting complexes and derived equivalence for self-injective algebras

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.

遗传阿贝尔范畴的周期导出Hall代数

Periodic derived Hall algebras of hereditary abelian categories

本文利用有界导出范畴的导出Hall数，为遗传阿贝尔范畴的周期导出范畴定义m-周期扩张导出Hall代数，从而统一显式刻画Bridgeland周期复形Hall代数及Xu-Chen奇周期导出Hall代数的代数结构。

关于分次温和单圈代数的导出Hall代数 II：Serre 表示

On the derived Hall algebra of a graded gentle one-cycle algebra II: Serre presentation

本文针对分次温和单圈代数，在积分不变量非0或±1时，给出了其导出Hall代数的Serre型生成元与关系。

关于温柔二圈代数的导出等价分类

On derived equivalence classification of gentle two-cycle algebras

我们分类，直到导出（等价地，倾斜cotilting）等价所有的非退化温柔的双循环代数。在退化情形下，我们给出了一个部分分类并提出了一个猜想。

国产型对称代数的导等价分类

Derived equivalence classification of symmetric algebras of domestic type

本文给出了代数闭域上所有国内表示型对称代数的一个完整的导出等价分类。这完成了R. Bocian和作者，在本文中，我们解决的关键问题，区分标准和非标准代数导出等价。我们的主要工具是由B. K“ulshammer关于正特征对称代数的一个新的证明，最近A. Zimmermann（数学。RA/0509580）在导出的等价物下是不变量。

Singularity categories of derived categories of hereditary algebras are derived categories

We show that for the path algebra $A$ of an acyclic quiver, the singularity category of the derived category $\mathsf{D}^{\rm b}(\mathsf{mod}\,A)$ is triangle equivalent to the derived category of the functor category of $\underline{\mathsf{mod}}\,A$, that is, $\mathsf{D}_{\rm sg}(\mathsf{D}^{\rm b}(\mathsf{mod}\,A))\simeq \mathsf{D}^{\rm b}(\mathsf{mod}(\underline{\mathsf{mod}}\,A))$. This extends a result of Iyama-Oppermann for the path algebra $A$ of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.

单参数自内射代数的导等价分类

Derived equivalence classification of one-parametric selfinjective algebras

给出了代数闭域上所有单连通Galois覆盖的单参数自内射代数的等价规范形的完整刻画。作为结果，这些代数的稳定等价规范形的描述也得到了。

Modified Ringel-Hall Algebras, Green's formula and Derived Hall Algebras

In this paper we define the modified Ringel-Hall algebra $\cm\ch(\ca)$ of a hereditary abelian category $\ca$ from the category $C^b(\mathcal{A})$ of bounded $\mathbb{Z}$-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hal…

导等价与有限维代数

Derived equivalences and finite dimensional algebras

讨论了有限维代数和有限群的表示论的同调代数。我们提出了构造和研究导出范畴等价性的各种方法：局部群论、几何和代数。

Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras II: braid group actions

Based on the realization of quantum Borcherds-Bozec algebra $\widetilde{\mathbf{U}}$ and quantum generalized Kac-Moody algebra ${}^B\widetilde{\mathbf{U}}$ via semi-derived Ringel-Hall algebra of a quiver with loops, we deduce the braid group actions of $\widetilde{\mathbf{U}}$ introduced by Fan and Tong recently and establish braid group actions for ${}^B\widetilde{\mathbf{U}}$ by applying the BGP reflection functors to semi-derived Ringel-Hall algebras.