Igor Burban, Yuriy Drozd
In this article we describe indecomposable objects of the derived categories of a branch class of associative algebras. To this class belong such known classes of algebras as gentle algebras, skew-gentle algebras and certain degenerations of tubular algebras.
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We give a result of Auslander-Ringel-Tachikawa type for Gorenstein-projective modules over a complete Gorenstein order. In particular, we prove that a complete Gorenstein order is of finite Cohen-Macaulay representation type if and only if every indecomposable pure-injective object in the stable category of Gorenstein-projective modules is compact.
The category ${\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\mathbb C^n$, \cite{JKS16}. Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\{1,2,\dots,n\}$, and their explicit construction has been given by Jensen, King and Su. These are the building blocks of the category as any module in ${\rm CM}(B_{k,n}) $ can be filtered by the…
Let k be an algebraically closed field and A a finite dimensional associative k-algebra. We prove that there is no gap in the lengths of indecomposable A-modules of finite length. The analogous result holds for an abelian k-linear category C if the endomorphism algebras of the simples are k.
Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories
该研究针对Khovanov-Sazdanovic提出的Deligne插值范畴的推广形式，通过引入带滤过Krull-Schmidt范畴的相伴分次范畴等工具，完成了不可分解对象的分类并确定了相伴分次Grothendieck环。
Rigid Indecomposable Modules in Grassmannian Cluster Categories
本文研究格拉斯曼簇范畴中刚性不可分解模与Kac-Moody根系的对应关系，证明秩2和秩3模均对应根，并确认秩3模轮廓数量，提出典型轮廓循环置换猜想。
On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type
本文证明了驯顺型cluster-tilted代数上不同的不可分解$τ$-刚性模具有不同的维数向量，并以此为基础建立了驯顺型cluster algebras的弱版本分母猜想，同时利用Iyama-Yoshino构造描述了相关的三角范畴subfactors。
Categories of generalized thread quivers
本研究聚焦线程箭图及其商的表示范畴，证明其逐点有限维表示的不可分解性诱导关系，分类内射投射不可分解表示，刻画相关范畴表示性质并构造新的遗传阿贝尔范畴。
We study Karoubian tensor categories which interpolate representation categories of families of so-called easy quantum groups in the same sense in which Deligne's interpolation categories $\mathrm{\underline{Rep}}(S_t)$ interpolate the representation categories of the symmetric groups. As such categories can be described using a graphical calculus of partitions, we call them interpolating partition categories. They include $\mathrm{\underline{Rep}}(S_t)$ as a special case and can generally be viewed as subcategories of the latter. Focusing on semisimplicity and descriptions of the indecomposab…
We study the homotopy category $ K(\Inj A)$ of all injective modules over a finite dimensional algebra $A$ with discrete derived category. We give a classification of the indecomposable objects of $ K(\Inj A)$ for any radical square zero self-injective algebra $A$. In particular, every indecomposable object is endofinite.
A correspondence between additive and monoidal categorifications with application to Grassmannian cluster categories
本文提出簇代数的加法与幺半范畴化的猜想对应，通过格拉斯曼簇范畴构造A型通用核，并应用该对应在格拉斯曼簇范畴中构造刚性与非刚性不可分解模。
Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely defined by its index with respect to $\mathscr{T}$. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angula…
We introduce the Calabi-Yau (CY) objects in a Hom-finite Krull-Schmidt triangulated $k$-category, and notice that the structure of the minimal, consequently all the CY objects, can be described. The relation between indecomposable CY objects and Auslander-Reiten triangles is provided. Finally we classify all the CY modules of self-injective Nakayama algebras, determining this way the self-injective Nakayama algebras admitting indecomposable CY modules. In particular, this result recovers the algebras whose stable categories are Calabi-Yau, which have been obtained in [BS].
By using the approach in \cite{XX2006} to Hall algebras arising in homologically finite triangulated categories, we find an `almost' associative multiplication structure for indecomposable objects in a 2-periodic triangulated category. As an application, we give a new proof of the theorem of Peng and Xiao in \cite{PX2000} which provides a way of realizing symmetrizable Kac-Moody algebras and elliptic Lie algebras via 2-periodic triangulated categories.
We derive an algorithm for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras coming from standard algebraic 2-CY triangulated categories with a finite number of indecomposables. These form a class of algebras that satisfy the setup for our mutation algorithm.
We determine the derived representation types of algebras with radical square zero and give a description of the indecomposable objects in their bounded derived categories.
C. Amiot has classified the connected triangulated k-categories with finitely many isoclasses of indecomposables satisfying suitable hypotheses. We remark that her proof shows that these triangulated categories are determined by their underlying k-linear categories. We observe that if the connectedness assumption is dropped, the triangulated categories are still determined by their underlying k-categories together with the action of the suspension functor on the set of isoclasses of indecomposables.
We classify the derived tame Schur and infinitesimal Schur algebras and describe indecomposable objects in their derived categories.
A functorial approach to monomorphism categories II: Indecomposables
本文通过构造单态范畴的内射稳定范畴到箭图表示范畴的 epivalence，并利用广义 Mimo-构造给出其逆，刻画了单态范畴的不可分解对象，进而应用于 Nakayama 代数、单列环以及奇点范畴和 ιquiver 代数等具体情形。
In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of arcs in the surface and give explicit…
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formed by inducing the simple modules of the maximal subgroups of $S$ via th…
In this article we classify indecomposable objects of the derived categories of finitely-generated modules over certain infinite-dimensional algebras. The considered class of algebras (which we call nodal algebras) contains such well-known algebras as the complete ring of a double nodal point $\kk[[x,y]]/(xy)$ and the completed path algebra of the Gelfand quiver. As a corollary we obtain a description of the derived category of Harish-Chandra modules over $SL_{2}({\mathbb R})$. We also give an algorithm, which allows to construct projective resolutions of indecomposable complexes. In the appen…
Let $A$ be a hereditary algebra. We construct a fundamental domain for the cluster category of $A$ inside the category of modules over the duplicated algebra $\bar{A}$ of $A$. We then prove that there exists a bijection between the tilting objects in the cluster category and the tilting $\bar{A}$-modules all of whose non projective-injective indecomposable summands lie in the left part of the module category of $\bar{A}$.
We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables. We use this to show that the global dimension of a finite dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.
Krull-Schmidt categories are additive categories such that each object decomposes into a finite direct sum of indecomposable objects having local endomorphism rings. We provide a self-contained introduction which is based on the concept of a projective cover.
We study endomorphism algebras of 2-term silting complexes in derived categories of hereditary finite dimensional algebras, or more generally of $\mathop{\rm Ext}\nolimits$-finite hereditary abelian categories. Module categories of such endomorphism algebras are known to occur as hearts of certain bounded $t$-structures in such derived categories. We show that the algebras occurring are exactly the algebras of small homological dimension, which are algebras characterized by the property that each indecomposable module either has injective dimension at most one, or it has projective dimension a…
The first Brauer-Thrall conjecture for extriangulated length categories
本文在外三角长度范畴中引入关于 $Θ$ 的 Gabriel-Roiter 测度并推广 Gabriel 性质，在技术性条件下证明第一 Brauer-Thrall 猜想。
We show that all strongly non-degenerate trigonometric solutions of the associative Yang-Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya category of square-tiled surfaces. Along the way, we give a classification result for cyclic $A_\infty$-algebra structures on a certain Frobenius algebra associated with a pair of 1-spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf. arXiv:1601.06141), we prove that any two simple vect…
We study Hom-type analogs of Rota-Baxter and dendriform algebras, called Rota-Baxter $G$-Hom-associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota-Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.
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…
We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form $\mathbb{Z}Δ/G$ where $Δ$ is a disjoint union of simply laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}Δ$. Then we prove that for `most' groups $G$, the category $\T$ is standard, \emph{i.e.} $k$-linearly equivalent to an orbit category $\mathcal{D}^b(\modd kΔ)…
We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.
In this survey, we first present basic facts on A-infinity algebras and modules including their use in describing triangulated categories. Then we describe the Quillen model approach to A-infinity structures following K. Lefevre's thesis. Finally, starting from an idea of V. Lyubashenko's, we give a conceptual construction of A-infinity functor categories using a suitable closed monoidal category of cocategories. In particular, this yields a natural construction of the bialgebra structure on the bar construction of the Hochschild complex of an associative algebra.
We develop the basic constructions of homological algebra in the (appropriately defined) unbounded derived categories of modules over algebras over coalgebras over noncommutative rings (which we call semialgebras over corings). We define double-sided derived functors SemiTor and SemiExt of the functors of semitensor product and semihomomorphisms, and construct an equivalence between the exotic derived categories of semimodules and semicontramodules. Certain (co)flatness and/or (co)projectivity conditions have to be imposed on the coring and semialgebra to make the module categories abelian (an…
We describe the modules in the Ziegler closure of ray and coray tubes in module categories over finite-dimensional algebras. We improve slightly on Krause's result for stable tubes by showing that the inverse limit along a coray in a ray or coray tube is indecomposable, so in particular, the inverse limit along a coray in a stable tube is indecomposable. In order to do all this, we first describe the finitely presented modules over and the Ziegler spectra of iterated one-point extensions of valuation domains. Finally we give a sufficient condition for the $k$-dual of a $Σ$-pure-injective modul…
Let G be a connected semisimple algebraic group over $k$, with Lie algebra $\g$. Let $\h$ be a subalgebra of $\g$. A simple finite-dimensional $\g$-module V is said to be $\h$-indecomposable if it cannot be written as a direct sum of two proper $\h$-submodules. We say that $\h$ is wide, if all simple finite-dimensional $\g$-modules are $\h$-indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of $\g$ and initiate their systematic study. Our approach is ba…
For an object $A$ of a category of interest $\mathbb{C}$ we construct the group with operations ${\mathfrak{B}}(A)$ and the semidirect product ${\mathfrak{B}}(A)\ltimes A$ and prove that there exists an actor of $A$ in $\mathbb{C}$ if and only if ${\mathfrak{B}}(A)\ltimes A \in \mathbb{C}$. The cases of groups, Lie, Leibniz, associative, commutative associative, alternative algebras, crossed and precrossed modules are considered. The paper contains some results for the case $Ω_2 = \{+,*,*^{\circ}\}$.
Marked non-orientable surfaces and cluster categories via symmetric representations
本文开创了非可定向曲面表示理论，通过对称表示提升至簇范畴，建立不可分解对称对象与曲面曲线及拟三角剖分的对应，为拟簇代数的范畴化奠定基础。
Flat model structures and Gorenstein objects in functor categories
本文构建了在泛函范畴上的平坦模型结构，并证明了在特定条件下，对象的戈尔登斯坦性质等价于其在每个对象上的性质，改进了已有结论。
We study semiorthogonal decompositions of bounded derived categories of gentle algebras and how they are manifested in the geometric model of these categories as constructed by Opper, Plamondon and Schroll. We prove that there is a one-to-one correspondence between such semiorthogonal decompositions and suitable cuts of the marked surface underlying the geometric model. Our main tool is the characterization of basis morphisms between indecomposable objects due to Arnesen, Laking and Pauksztello.
For a finite group $G$ and an algebraically closed field $k$ of characteristic $p>0$ for any indecomposable finite dimensional $kG$-module $M$ with vertex $D$ and a subgroup $H$ of $G$ containing $N_G(D)$ there is a unique indecomposable $kH$-module $N$ of vertex $D$ being a direct summand of the restriction of $M$ to $H$. This correspondence, called Green correspondence, was generalised by Auslander-Kleiner to the situation of pairs of adjoint functors between additive categories. In the original situation of group rings Carlson-Peng-Wheeler proved that this correspondence is actually restric…
Representation Theory of $0$-Schur Algebras and Related Categories
本文针对0-Schur代数$\mathbf{S}_0(n,r)$，通过计算其投射不可分解模的顶识别单模，并考察其与0-Hecke代数、退化量子群对应模范畴间的函子关系。
A graph theoretic model for the derived categories of gentle algebras and their homological bilinear forms
为温和代数的有界导出范畴定制图论模型，用于分析不可分解对象、Auslander-Reiten三角及其与同调双线性型和Coxeter变换的相互作用，并探索新的导出不变量。
For a suitable triangulated category $\mathcal{T}$ with a Serre functor $S$ and a full precovering subcategory $\mathcal{C}$ closed under summands and extensions, an indecomposable object $C$ in $\mathcal{C}$ is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there is no Auslander-Reiten triangle in $\mathcal{C}$ with end term $C$. In this paper, we show that if, for such an object $C$, there is a minimal right almost split morphism $β:B\rightarrow C$ in $\mathcal{C}$, then $C$ appears in something very similar to an Auslander-Reiten triangle in $\mathcal{C}$: an essentially unique tri…
We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with finitely-generated homogeneous components are shown to be isomorphic to direct sums of indecomposable string and band complexes. The corresponding isoclasses are described, and the Krull-Remak-Schmidt-Azumaya property is verified. This classification problem is solved using the idea of functorial filtrations.
Type $\textrm{II}$ quantum subgroups for quantum $\mathfrak{sl}_N$. $\textrm{II}$: Classification
本文分类了仿射 Kac-Moody $\mathfrak{sl}_N$ 可积表示范畴上的不可分解模范畴，在一般 $k$ 情形给出相对张量积公式，并在 $N\leq 7$ 的非一般情形发现例外模范畴。
Triangulated structure for Bondarenko's Categories
本文针对 Bondarenko 范畴非阿贝尔但与 gentle 代数导出范畴密切相关的问题，在其商范畴上构造了三角范畴结构，为运用三角范畴技术研究偏序集表示提供了新工具。
Classification of objects in the singularity categories of rational double points in arbitrary characteristics
本文研究了任意特征代数闭域上的有理二重点，完全分类了其奇点范畴中的不可分解对象，对应于Auslander-Reiten箭图中的顶点，并给出了确定这些箭图配置的替代证明以及处理孤立超曲面奇点矩阵因子化同伦范畴的计算机代数方法。
We continue the study of the lower central series of a free associative algebra, initiated by B. Feigin and B. Shoikhet (arXiv:math/0610410). We generalize via Schur functors the constructions of the lower central series to any symmetric tensor category; specifically we compute the modified first quotient \bar{B}_1, and second and third quotients B_2, and B_3 of the series for a free algebra T(V) in any symmetric tensor category, generalizing the main results of (arXiv:math/0610410) and (arXiv:0902.4899). In the case A_{m|n}:=T(\CC^{m|n}), we use these results to compute the explicit Hilbert s…
Fukaya categories of orbifold surfaces in representation theory
本文通过轨道面的Fukaya范畴研究，揭示了轨道表面与斜gentle代数之间的导出等价关系，并利用变形理论探讨其联系。
We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.
We prove that the bounded derived category of the incidence algebra of the Tamari lattice is fractionally Calabi-Yau, giving a positive answer to a conjecture of Chapoton. The proof involves a combinatorial description of the Serre functor of this derived category on a sufficiently nice family of indecomposable objects.
We construct a differential graded algebra (DGA) modelling certain $A_\infty$ algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely $H^*BG$ and $H_*ΩBG^{^\wedge}_p$. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander--Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.
The indecomposable objects in the center of Deligne's category $Rep(S_t)$
本研究将$Rep(S_t)$视为模型论极限，分类了德利涅范畴$Rep(S_t)$幺半中心的不可分解对象，证明其中心半单等价于$t$非非负整数，还等同了相关Grothendieck环并得到阿贝尔包络的类似结果。
For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable monomorphism category of the module category of $A$…
We study maximal $m$-rigid objects in the $m$-cluster category $\mathcal C_H^m$ associated with a finite dimensional hereditary algebra $H$ with $n$ nonisomorphic simple modules. We show that all maximal $m$-rigid objects in these categories have exactly $n$ nonisomorphic indecomposable summands, and that any almost complete $m$-rigid object in $\mathcal C_H^m$ has exactly $m+1$ nonisomorphic complements. We also show that the maximal $m$-rigid objects and the $m$-cluster tilting objects in these categories coincide, and that the class of finite dimensional algebras associated with maximal $m$…
We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.
We prove that the weak associativity for modules for vertex algebras are equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requiring that these two formulas instead of the commutator formula hold, we construct a functor $S$ from the category of modules for Zhu's algebra of a vertex operator algebra $V$ to the category of $\mathbb{N}…
We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type A) we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.
In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and tau-tilting theory in module categories. Our main results show that relative rigid objects are in bijection with $τ$-rigid pairs, and also relative maximal rigid objects with support tau-tilting pairs under some assumptions. These results generalize their work by Adachi-Iyama-Reiten, Yang-Zhu and Fu-Geng-Liu. Finally, we introduce mutation of relative maximal rigid objects and show that any basic relative almost maximal rigid object has exactly two non-isomorphic indecomposable…
In this paper, we investigate properties of the bounded derived category of finite dimensional modules over a gentle or skew-gentle algebra. We show that the Rouquier dimension of the derived category of such an algebra is at most one. Using this result, we prove that the Rouquier dimension of an arbitrary tame projective curve is equal to one, too. Finally, we elaborate the classification of indecomposable objects of the (possibly unbounded) homotopy category of projective modules of a gentle algebra.
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