Pierre-Guy Plamondon
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.
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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
Tilting-completion for gentle algebras
本文利用曲面模型和曲面切割归纳法，证明了gentle代数上的几乎倾斜模均为部分倾斜模且可完备化，至多有$2n$个补模，从而证实了Happel的猜想，并构造了非部分倾斜的预倾斜模反例。
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。
On $τ$-tilting modules over trivial extensions of gentle tree algebras
本研究刻画了gentle树代数平凡扩张的结构（对应无例外顶点的Brauer树代数等），并得出其支撑$τ$-倾斜模数量仅由单模数量决定的结论。
On the $τ$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras
本文研究了带权（非对称）温和代数的τ-倾斜有限性和硅定离散性，证明了其与表示有限性的等价性，并通过几何方法刻画了硅定离散性的条件。
We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra $Λ$ contains a $d$-cluster tilting subcategory for some $d \geq 2$, then $Λ$ is a radical square zero Nakayama algebra. This gives a complete classification of weakly $d$-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if $\mathcal{D}^b(Λ)$ contains a $d$-cluster tilting subcategory that is clos…
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。
Resolving subcategories for gentle algebras III: Tilting modules for gentle tree algebras
本文通过改进的曲面模型，为有限整体维数的温和树代数开发了组合、偏序集和箭图表示技术，实现了解决子范畴与倾斜模之间的Auslander-Reiten一一对应的组合实现。
Tilting mutation of gentle algebras and its combinatorial description
本文通过广义 BB-倾斜模研究 gentle 代数的倾斜突变，给出基于箭向和关系的判据，并构造突变后自同态代数的组合描述，证明所得代数仍为 gentle 且导出等价。
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,τY$). We use this to describe support $τ$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $τ$-tilting modules and functorially finite torsion classes. We give an explicit basis of Ext$^1(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.
Let $\mathcal{D}$ be a Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with a rigid object $R$. Let $Λ=\operatorname{End}_{\mathcal{D}}R$ be the endomorphism algebra of $R$. We introduce the notion of mutation of maximal rigid objects in the two-term subcategory $R\ast R[1]$ via exchange triangles, which is shown to be compatible with mutation of support $τ$-tilting $Λ$-modules. In the case that $\mathcal{D}$ is the cluster category arising from a punctured marked surface, it is shown that the graph of mutations of support $τ$-tilting $Λ$-modules is isomorphic to the graph of fli…
On support $τ$-tilting graphs of gentle algebras
本文研究了代数闭域上有限维gentle代数的支撑$τ$-倾斜图的组合性质，证明了其连通性及面内可达性质，印证了cluster代数交换图的相关猜想。
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,…
Non-kissing complexes and tau-tilting for gentle algebras
本文为温和代数建立了支持τ-倾斜复形与非亲吻复形的对应，并证明了有限情形下递增翻转图构成同余均匀格，其g-向量扇为非亲吻关联多面体的法扇。
We introduce higher gentle algebras. Our definition allows us to determine the singularity categories and subsequently show that higher gentle algebras are Iwanaga-Gorenstein. Under extra assumptions, we show that cluster-tilted algebras (in the sense of Oppermann-Thomas) of higher Auslander algebras of type $A$ are higher gentle.
A Characteristic free approach to skew-gentle algebras
本文提出一种特征无关方法，通过建立同调满态射和导出模范畴的recollement关联skew-gentle代数与gentle代数，解决了自injective分类等多个开放问题并加强了已知猜想。
The algebras derived equivalent to gentle cluster tilted algebras
已知一个簇倾斜代数是缓和的当且仅当它是Dynkin型$\bbA$或欧几里得型$\tilde{\bbA}$的簇倾斜代数。我们将有限维代数分类为等价于温柔簇倾斜代数的有限维代数。
Tilting mutations as generalized Kauer moves for (skew) Brauer graph algebras with multiplicity
本文将广义Kauer移动与倾斜突变诱导导出等价的结果推广到任意重数Brauer图代数及斜Brauer图代数，并证明重数为1的斜Brauer图代数是斜gentle代数的平凡扩张。
We describe, in terms of generators and relations, the derived Hall algebras associated to the one-cycle gentle algebras of infinite global dimension.
Brick-finite skew-gentle algebras are representation-finite
本文证明斜有限非正则代数与表示有限性等价，扩展了Plamondon关于温和代数的结果。
In this paper we define almost gentle algebras. They are monomial special multiserial algebras generalizing gentle algebras. We show that the trivial extension of an almost gentle algebra by its minimal injective co-generator is a symmetric special multiserial algebra and hence a Brauer configuration algebra. Conversely, we show that any almost gentle algebra is an admissible cut of a unique Brauer configuration algebra and as a consequence, we obtain that every Brauer configuration algebra with multiplicity function identically one, is the trivial extension of an almost gentle algebra. We sho…
Hochschild cohomology of gentle algebras
计算了温柔代数的Hochschild上同调群，证明了它们是由Avella-Alaminos和Geiss引入的导出不变量决定的.
The almost split triangles for perfect complexes over gentle algebras
本文描述了温柔代数上完全复形同伦范畴中的几乎可裂序列。
The derived equivalence classification of gentle two-cycle algebras
我们完成了由Avella-Alaminos和Bobinski-Malicki在早期论文中提出的温柔双圈代数的导出等价分类。
Homological conditions on locally gentle algebras
本文研究局部gentle代数的同调性质，通过组合方法描述其中心、素谱和同调维数，并分类其Artin-Schelter Gorenstein、正则及Cohen-Macaulay条件，推广了Stanley定理。
Homological dimensions over almost gentle algebras
本文提出计算几乎温和代数全局维数与自内射维数的方法，并证明满足Auslander条件的此类代数为Gorenstein代数。
In this paper we determine the first Hochschild homology and cohomology with different coefficients for gentle algebras and we give a geometrical interpretation of these (co)homologies using the ribbon graph of a gentle algebra as defined in earlier work by the second author. We give an explicit description of the Lie algebra structure of the first Hochschild cohomology of gentle and Brauer graph algebras (with multiplicity one) based on trivial extensions of gentle algebras and we show how the Hochschild cohomology is encoded in the Brauer graph. In particular, we show that except in one low-…
An extended version of Riedtmann's Lie algebra of the gentle one-cycle algebra $Λ(n-1,1,1)$ is computed and is shown to admit a Cartan decomposition by the positive roots of the root system of type $\mathbf{BC}_n$.
On indecomposable exceptional modules over gentle algebras
给出了有限维温柔代数上不可分解例外模的一个刻划。作为应用，我们研究了由非穿孔曲面生成的温柔代数，并证明了一类与无自交曲线相关的不可分解模作为例外模，是由它们的维数向量唯一确定的.
We give a complete description of a basis of the extension spaces between indecomposable string and quasi-simple band modules in the module category of a gentle algebra.
A geometric model for semilinear locally gentle algebras
本文推广温和代数为半线性局部温和代数，证明其结点性，并通过曲面切割与粘合给出有限维模的几何模型。
Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra
在这篇文章中，我们提供了一个简单的组合描述之间的不可分解的复合物的有界导出范畴的温柔代数的态射。
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。
This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (19-24 January 2020). It introduces a geometric orbifold model for the bounded derived category of a skew-gentle algebra. This is joint work with Daniel Labardini-Fragoso and Yadira Valdivieso-Díaz.
Trivial Extensions of Gentle Algebras and Brauer Graph Algebras
我们表明，两个良好的研究类的驯服代数一致：即，对称特殊bisling-algebra代数类的Brauer图代数一致。然后，我们探讨温柔代数和对称特殊双代数之间的联系，明确确定平凡扩张的温柔代数的最小内射的共同生成元。这是一个对称的特殊双曲代数，因此是一个Brauer图代数，我们明确给出Brauer图。我们进一步表明，布劳尔图代数产生，通过允许的削减，许多温柔代数和平凡的扩张通过一个允许的削减得到的温柔代数是原来的布劳尔图代数。作为结果，我们证明了边界上有标记点的Riemann曲面的理想三角剖分的Jacobian代数的平凡扩张同构于由三角剖分的弧给出Brauer图的Brauer图代数。
There are no strictly shod algebras in hereditary gentle algebras
本文利用几何模型证明遗传温和代数中不存在严格shod代数，并据此对Dynkin型A_n和Ã_n的silted代数进行了分类。
Singularity categories of skewed-gentle algebras
设$K$是代数闭域。设$(Q,Sp,I)$是一个斜缓三元组，$(Q^{sg},I^{sg})$和$(Q^g,I^{g})$分别是其对应的斜缓对和相伴的缓对.证明了斜柔代数$KQ^{sg}/< I^{sg}>$与$KQ/< I>$是奇异等价的.此外，我们使用$(Q,Sp,I)$来描述$KQ^g/< I^g>$的奇异性范畴。作为推论，我们得到$\mathrm{gldim} KQ^{sg}/< I^{sg}><\infty$当且仅当$\mathrm{gldim} KQ/< I><\infty$当且仅当$\mathrm{gldim} KQ^{g}/< I^{g}><\infty$。
Thick subcategories of derived categories of gentle algebras
本研究探讨温和代数导出范畴的厚子范畴，发现其由弦对象或带对象生成，并证明弦对象生成的厚子范畴与导出范畴几何模型的非交叉路径集存在双射对应。
Combinatorial derived invariants for gentle algebras
我们定义导出的等价不变量温柔代数，构造在一个简单的组合的方式，从定义这些代数的关系，从图中。我们的不变量由成对的自然数组成，并包含有关代数及其重复代数的稳定Auslander-Reiten代数的结构的重要信息。作为一个副产品，我们得到了一个温柔代数的矢的数量是不变的导出等价。最后，我们的不变量分离的温柔代数的导出等价类与最多一个循环。
Desingularization of quiver Grassmannians for Gentle algebras
继[20]之后，利用Cohen-Macaulay Auslander代数上的有限维Gorenstein投射模的任意非奇异Grassmannian，构造了1-Gorenstein温柔代数上的有限维Gorenstein投射模的非奇异Grassmannian.
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.
Skew-gentle algebras are skew-group algebras of certain gentle algebras endowed with a Z 2-action. Using the topological description of Opper, Plamondon and Schroll in [OPS] for the indecomposable objects of the derived category of any gentle algebra, one obtains here a complete description of indecomposable objects in the derived category of any skew-gentle algebras in terms of curves on an orbifold surface.The results presented here are complementary to the ones in [LSV]. First, we obtain a complete classification of indecomposable objects and not of ''homotopy strings'' and ''homotopy bands…
Derived classification of gentle algebras with two cycles
我们分类温柔代数定义的箭图下导出等价的两个循环在非退化的情况下，通过使用一些组合不变量构造的关系，定义这些代数的箭图。我们还提出了一个正规形式的列表;任何这样的代数导出等价于列表中的代数之一。
For any gentle algebra $Λ=KQ/\langle I\rangle$, following Kalck, we describe the quiver and the relations for its Cohen-Macaulay Auslander algebra $\mathrm{Aus}(\mathrm{Gproj}Λ)$ explicitly, and obtain some properties, such as $Λ$ is representation-finite if and only if $\mathrm{Aus}(\mathrm{Gproj}Λ)$ is; if $Q$ has no loop and any indecomposable $Λ$-module is uniquely determined by its dimension vector, then any indecomposable $\mathrm{Aus}(\mathrm{Gproj}Λ)$-module is uniquely determined by its dimension vector.
Recollements for graded gentle algebras from spherical band objects
本文研究梯度温和代数的导出范畴的局部化，将其描述为重收集，并引入梯度捏合温和代数，证明其与具有锥奇点的梯度标记曲面一一对应，局部化对应闭曲线收缩。
We classify finitely generated modules over a class of algebras introduced in the authors' Ph.D thesis, called complete gentle algebras. These rings generalise the finite-dimensional gentle algebras introduced by Assem and Skowroński, in such a way so that the ground field is replaced by any complete local noetherian ring. Our classification is written in terms of string and band modules. For the proof we apply the main result from the authors' thesis, which classifies complexes of projective modules with finitely generated homogeneous components up to homotopy. In doing so we construct the re…
On the derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure
本文在温和条件下将分级温和单圈代数的两类导出范畴描述为无限箭图的扭曲根范畴，并证明其三角结构由底层加性范畴唯一决定。
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.
On the invariant theory for acyclic gentle algebras
本文证明了无圈温柔代数的模簇的不可约分支上的有理不变式域是纯超越扩张。一路上，我们表现出这样的领域的有理不变式的超越基础Schofield行列式半不变式。我们还证明了正则不可约分支上模的模空间是射影空间的乘积。
Stable endomorphism algebras of modules over special biserial algebras
证明了一类特殊双曲代数上无自扩张模的稳定自同态代数是温柔代数。特别是，它又是特殊的比萨饼。因此，任何一个导出的代数等价于一个温柔代数，它就是温柔代数。
On the Ringel--Hall algebra of the gentle one-cycle algebra $Λ(n-1,1,1)$
本文证明温和单圈代数 $\Lambda(n-1,1,1)$ 具有 Hall 多项式，并显式给出所有不可分解模三元组的 Hall 多项式，进而证明其 Ringel--Hall 李代数与 Riedtmann 李代数同构。
Geometric models for the algebraic hearts in the derived category of a gentle algebra
本文通过变形并嵌入温和代数的两种几何模型，在带标记曲面上实现了导出范畴中任意代数心的几何刻画，并将高阶Yoneda-扩张与Yoneda-积分别实现为多边形及其粘合。
The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras
本文首次完整计算了温和代数族的Tamarkin-Tsygan演算，结合曲面几何模型解释了其与辛上同调的关联，并给出由该演算恢复温和代数结构结果的方法。
Quadratic Differentials as Stability Conditions of Graded Skew-gentle Algebras
本文证明分次斜温和代数心交换图的主分支等同于S-图交换图，并进一步建立稳定性条件空间与二次微分空间的同构。
On the Mutations of Gentle Quivers and Admissible Ideals
本文将Fomin-Zelevinsky箭图突变框架推广至带关系的箭图，通过在路径代数可许理想上引入对合突变操作，构建生成可许温和有界箭图代数的系统框架，并提出相关未来研究方向。
Gentle algebras arising from surface triangulations
本文将代数A（T）与曲面S的三角剖分T联系起来，曲面S具有一组边界标志点。这个代数A（T）是温柔的，是一维的Gorenstein。我们还证明了A（T）是簇倾斜的当且仅当它是A型或A型波浪线的簇倾斜的，或者当且仅当曲面S是圆盘或环面。此外，还得到了所有A型或A代字号的簇倾斜代数。
Derived invariants of gentle orders
针对无限维温和序，本文分解导出 Nakayama 函子、研究分数 Calabi–Yau 对象与例外循环，并证明基础箭图的若干组合不变量在导出等价下保持不变。
For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of strings, of the corresponding support $τ$-tilting module (or equivalently, two-term silting complexes). In the topological interpretation of gentle algebras via marked surfaces, such a set can be interpreted as a dissection (or partial triangulation), or equivalently, a lamination that does not contain a closed curve. We will refine this combinatorics, w…
Semi-Invariants for Gentle String Algebras
本文给出了一个算法，用于确定温柔弦代数的表示空间的不可约分支上的半不变函数环的生成元和关系。这些半不变量的环是半群环，我们可以将其与所谓的匹配图联系起来。在这种关联下，半群的生成元可以通过图上的某些行走来看到，并且关系可以通过图中的某些配置来给出。这使我们能够确定这些环的生成元和关系的度界。我们进一步表明，这些界限也持有一般的非循环弦代数。
A geometric model for the module category of a skew-gentle algebra
本文将斜温柔代数实现为带孔标记曲面容许部分三角剖分关联的斜平铺代数，建立几何对象与模范畴的双射，并据此分类支撑 $τ$-倾斜模。
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