Karin Baur, Bethany Marsh
We show that the m-cluster category of type D_n is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm-m+1-gon. This generalises a result of Schiffler for m=1. We use the notion of the mth power of a translation quiver to realise the m-cluster category in terms of the cluster category.
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We give a geometric realization of cluster categories of type $D_n$ using a polygon with $n$ vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
We show that the $m$-cluster category of type $A_{n-1}$ is equivalent to a certain geometrically-defined category of diagonals of a regular $nm+2$-gon. This generalises a result of Caldero, Chapoton and Schiffler for $m=1$. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the $m$th power of a translation quiver and show how it can be used to realise the $m$-cluster category in terms of the cluster category.
On geometrically reductive tensor categories
本文证明了高阶Verlinde范畴是几何约化的猜想，并进一步将关于张量范畴几何约化的两个猜想简化为文献中已有的猜想。
The aim of this note is to answer several open problems arising from the geometric description of the $m$-cluster categories of type $A_n$ and their realization in terms of the $m$-th power of a translation quiver. In particular, we give a geometric interpretation of the triangulated structure of $m$-cluster categories. Furthermore, we characterize all the connected components arising from a cluster category when taking the $m$-th power of its Auslander-Reiten quiver.
We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.
On geometric models in representation theory
本文探讨了几何模型在表示论中的应用，特别是温和代数的表面模型对导出范畴和Fukaya范畴的贡献。
In this paper we give a geometric-combinatorial description of the cluster categories of type E. In particular, we give an explicit geometric description of all cluster tilting objects in the cluster category of type E_6. The model we propose here arises from combining two polygons, and it generalises the description of the cluster category of type A and D.
The main result here gives an algebra(/linear category) isomorphism between a geometrically defined subcategory $J^1_0$ of a short Brauer category $J_0$ and a certain one-parameter specialisation of the blob category $b$. That is, we prove the Conjecture in Remark 6.7 of [14]. We also define a sequence of generalisations $J^i_{i-1}$ of the category $J^1_0$. The connection of $J_0$ with the blob category inspires a search for connections also with its beautiful representation theory. Here we obtain formulae determining the non-semisimplicity condition (generalising the classical `root-of-unity'…
The Soergel category B of a Coxeter system (W,S) is a bimodule category over a polynomial algebra on which W acts. It's a categorification of the Hecke Algebra of (W,S). In this article we give a combinatorial description of morphism spaces in B. As a corollary, we give an analogous description of the morphisms in O_0-proj, where O_0 is the principal block of the BGG category O. ----- La catégorie B de Soergel d'un système de Coxeter (W,S) est une catégorie de bimodules sur une algèbre de polynômes sur laquelle W agit. C'est une catégorification de l'algèbre de Hecke de (W,S). Dans cet article…
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field, and let $\mathcal{C}_{H}$ be the corresponding cluster category. We give a description of the (standard) fundamental domain of $\mathcal{C}_{H} $ in the bounded derived category $\mathcal{D}^{b}(H)$, and of the cluster-tilting objects, in terms of the category $\modΓ$\ of finitely generated modules over a suitable tilted algebra $% Γ.$ Furthermore, we apply this description to obtain (the quiver of) an arbitrary cluster-tilted algebra.
In this note we explain how to obtain cluster algebras from triangulations of (punctured) discs following the approach of S. Fomin, M. Shapiro and D. Thurston. Furthermore, we give a description of m-cluster categories via diagonals (arcs) in (punctured) polygons and of m-cluster categories via powers of translation quivers as given in joint work with R. Marsh.
We derive a geometric model for the module category $\operatorname{mod} \mathbf{k} Q$ of a Dynkin quiver $Q$ via the heart of a total stability condition on the bounded derived category of $\operatorname{mod} \mathbf{k} Q$. As an application, we prove Reineke's conjecture that there is a stability function on $\operatorname{mod} \mathbf{k} Q$ making any indecomposables stable.
The Dolbeault geometric Langlands conjecture via limit categories
该研究引入光滑栈余切栈的极限范畴，发展其理论并探讨与 D -模范畴关系，提出多贝尔几何朗兰兹猜想精确表述，证明极限范畴半正交分解，构造赫克算子，预期其蕴含半稳定希格斯丛 BPS 范畴等价，是拓扑镜像对称猜想的范畴版本。
Quantum Geometric Langlands Categories from N = 4 Super Yang-Mills Theory
本文利用导出代数几何和全纯陈-西蒙斯理论描述 N = 4 超杨-米尔斯理论的超对称扭转族，并通过范畴几何量子化给出量子几何朗兰兹对应中扭转 D-模范畴的量子场论综合。
We show that a subcategory of the $m$-cluster category of type $\tilde{D_n}$ is isomorphic to a category consisting of arcs in an $(n-2)m$-gon with two central $(m-1)$-gons inside of it. We show that the mutation of colored quivers and $m$-cluster-tilting objects is compatible with the flip of an $(m+2)$-angulation. In the final part of this paper, we detail an example of a quiver of type $\tilde{D_7}$.
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented qu…
We study the Wakimoto modules over the affine Kac-Moody algebras at the critical level from the point of view of the equivalences of categories proposed in our previous works, relating categories of representations and certain categories of sheaves. In particular, we describe explicitly geometric realizations of the Wakimoto modules as Hecke eigen-D-modules on the affine Grassmannian and as quasi-coherent sheaves on the flag variety of the Langlands dual group.
We provide several equivalent descriptions of a highest weight category using recollements of abelian categories. Also, we explain the connection between sequences of standard and exceptional objects.
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.
Geometric models for the derived categories of Ginzburg algebras of n-angulated surfaces via local-to-global principles
本文通过局部到整体原理，用曲面曲线和相交构造相对Ginzburg代数导出范畴的几何模型，并在n=3时匹配Ext群与簇代数扩展突变矩阵。
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.
The geometric Satake correspondence provides an equivalence of categories between the Satake category of spherical perverse sheaves on the affine Grassmannian and the category of representations of the dual group. In this note, we define a combinatorial version of the Satake category using irreducible components of fibres of the convolution morphism. We then prove an equivalence of coboundary categories between this combinatorial Satake category and the category of crystals of the dual group.
We revisit the geometric description of cluster categories in type E in terms of colored diagonals in a polygon and generalize it to the case of m-cluster categories. As an application, we relate colored diagonals in a polygon to semi-standard Young tableaux, in type E_6,E_7,E_8. This provides a new compatibility description of semi--standard Young tableaux in Grassmannian cluster algebras in type E_6, E_8 and in a sub-cluster algebra of type E_7.
We give a complete classification of (co)torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects which are not cluster tilting. These finite $2$-Calabi-Yau triangulated categories are divided into two main classes: one denoted by $\mathcal{A}_{n,t}$ called of type $A$, and the other denoted by $D_{n,t}$ called of type $D$. By using the geometric model of cluster categories of type $A, $ or type $D$, we give a geometric description of (co)torsion pairs in $\mathcal{A}_{n,t}$ or $D_{n,t}$ respectively, via defining the periodic Ptolemy diagrams. This allows to c…
The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent to an ideal quotient of a subcategory of C. On the other hand, Aslak Buan and Robert Marsh proved that this module category is also equivalent to some localisation of C. In this paper, we give a conceptual interpretation, inspired from homotopical algebra, of this double description. Our main aim, yet to be acheived, is to generalise Buan-Marsh's result to the case o…
We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra. For each triangulated orbit category T we give an explicit description of a representation-finite self-injective standard algebra with stable module category triangle equivalent to T.
We construct continuous Frobenius categories of type $D$. The stable categories of these Frobenius categories are cluster categories which contain the standard cluster categories of type $D_n$. When $n=\infty$, maximal compatible sets of indecomposable objects are laminations of the punctured disk. Discrete laminations are clusters. This new construction is topological and it also gives an algebraic interpretation of the "tagged arcs" which occur in Schiffler's geometric description of clusters of type $D_n$.
For each integer $k\geq 4$ we describe diagrammatically a positively graded Koszul algebra $\mathbb{D}_k$ such that the category of finite dimensional $\mathbb{D}_k$-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type ${\rm D}_k$ or ${\rm B}_{k-1}$, constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra. Properties like graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.
A geometric model for blocks of Frobenius kernels
本文基于 Steinberg 张量积公式的几何对应，定义并研究了 Langlands 对偶群的仿射 Grassmannian 上的反常层范畴，为代数群 Frobenius 核的表示块提供几何模型，并证明其具有射影、内射对象且为最高权范畴。
In this article, we study the geometric realizations of $m$-cluster categories of Dynkin types A, D, $\tilde{A}$ and $\tilde{D}$. We show, in those four cases, that there is a bijection between $(m+2)$-angulations and isoclasses of basic $m$-cluster tilting objects. Under these bijections, flips of $(m+2)$-angulations correspond to mutations of $m$-cluster tilting objects. Our strategy consists in showing that certain Iyama-Yoshino reductions of the $m$-cluster categories under consideration can be described in terms of cutting along an arc the corresponding geometric realizations. This allows…
By a local geometric Langlands correspondence for a complex reductive group G we understand a construction which assigns to a local system on the punctured disc for the Langlands dual group of G, a category equipped with an action of the formal loop group G((t)). We propose a conjectural description of these categories as categories of representations of the corresponding affine Kac-Moody algebra of critical level, and, in some cases, as categories of D-modules on the ind-schemes G((t))/K. We describe in detail these categories and interrelations between them and provide supporting evidence fo…
We consider categorical and geometric purity for sheaves of modules over a scheme satisfying some mild conditions, both for the category of all sheaves and for the category of quasicoherent sheaves. We investigate the relations between these four purities and compute a number of examples, in particular describing both the geometric and categorical Ziegler spectra for the category of quasicoherent sheaves over the projective line over a field.
We prove a weak version of Lusztig's conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine), and explain its relation to Lusztig's classification of character sheaves.
This paper establishes a link between the theory of cluster algebras and the theory of representations of partially ordered sets. We introduce a class of posets by requiring avoidance of certain types of peak-subposets and show that these posets can be realized as the posets of quivers of type $\mathbb{A}$ with certain additional arrows. This class of posets is therefore called \emph{posets of type $\mathbb{A}$}. We then give a geometric realization of the category of finitely generated socle-projective modules over the incidence algebra of a poset of type $\mathbb{A}$ as a combinatorial categ…
We introduce the idea of a geometric categorical Lie algebra action on derived categories of coherent sheaves. The main result is that such an action induces an action of the braid group associated to the Lie algebra. The same proof shows that strong categorical actions in the sense of Khovanov-Lauda and Rouquier also lead to braid group actions. As an example, we construct a braid group action on derived categories of coherent sheaves on cotangent bundles to partial flag varieties.
A proof of Dolbeault geometric Langlands for $\mathrm{GL}_2$ with reduced spectral curves
本文证明了对 $\mathrm{GL}_2$ 在谱曲线约化的 Hitchin 基上的 Dolbeault 几何 Langlands 对应，引入极限范畴作为经典极限，并连接了范畴 Donaldson-Thomas 理论与几何 Langlands 对应。
Derived categories of character sheaves II: canonical induction/restriction functors
本文推广了特征层 dg 范畴的组合描述至一般复约化群，并明确了抛物诱导/限制函子。
Perverse schobers are categorifications of perverse sheaves. We construct a perverse schober on a partial compactification of the stringy Kähler moduli space (SKMS) associated by Halpern-Leistner and Sam to a quasi-symmetric representation X of a reductive group G, extending the local system of triangulated categories established by them. The triangulated categories appearing in our perverse schober are subcategories of the derived category of the quotient stack X/G.
Polynomial functors on flags
本文推广了向量空间标志范畴上的 Schur 函子，利用组合范畴与无限秩群的表示刻画该函子范畴，并应用于研究一般线性群抛物子群的多项式表示及表示稳定性。
We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation theory. Our results rest on the notion of polynomial functor introduced by Eilenberg and Mac Lane. We give applications to representations of general linear groups or to finiteness properties of functor categories.
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。
Mutation of $n$-cotorsion pairs in triangulated categories
本文在三角范畴中定义了推广经典余挠对的$n$-余挠对概念，证明了其突变仍为$n$-余挠对，推广了Zhou和Zhu的工作，并给出了$A$型$n$-丛范畴中$n$-余挠对及其突变的几何刻画与实现。
This paper builds on earlier work, where the authors described Whittaker modules for the Virasoro algebra. Using a framework of Batra and Mazorchuk, the current paper investigates a category of Virasoro algebra modules that includes Whittaker modules. Results in this paper include a classification of the simple modules in the category and a description of certain induced modules that are a natural generalization of simple Whittaker modules.
We study the following generalization of singularity categories. Let X be a quasi-projective Gorenstein scheme with isolated singularities and A a non-commutative resolution of singularities of X in the sense of Van den Bergh. We introduce the relative singularity category as the Verdier quotient of the bounded derived category of coherent sheaves on A modulo the category of perfect complexes on X. We view it as a measure for the difference between X and A. The main results of this thesis are the following. (i) We prove an analogue of Orlov's localization result in our setup. If X has isolated…
A geometric construction of Lusztig's modified quantum algebra of symmetric type is presented by using certain localized equivariant derived categories of double framed representation varieties of quivers.
A geometric realization of socle-projective categories for posets of type $\mathbb{D}$
本文引入类型 $\mathbb{D}$ 偏序集，证明其socle-投射范畴有限表示型，并通过带孔多边形弧范畴与模范畴的等价给出几何实现，空alien箭头时cluster子代数等于完整cluster代数。
Leclerc recently studied certain Frobenius categories in connection with cluster algebra structures on coordinate rings of intersections of opposite Schubert cells. We show that these categories admit a description as Gorenstein projective modules over an Iwanaga-Gorenstein ring of virtual dimension at most two. This is based on a Morita type result for Frobenius categories.
Geometric Points in Tensor Triangular Geometry
本文研究张量三角几何中的几何点，通过高阶扎里斯基几何中的自由构造构造反例，讨论构造谱，并利用这些谱构建几何 incarnations of 原子素。
Affine W-algebras are a somewhat complicated family of (topological) associative algebras associated with a semisimple Lie algebra, quantizing functions on the algebraic loop space of Kostant's slice. They have attracted a great deal of attention in geometric representation theory because of Feigin-Frenkel's duality theorem for them, which identifies W-algebras for a Lie algebra and its Langlands dual through a subtle construction. The main result of this paper is an affine version of Skryabin's theorem, describing the category of modules over the W-algebra in simpler categorical terms. But un…
Realising perfect derived categories of Auslander algebras of type A as Fukaya-Seidel categories
本文证明了特定类型A的Auslander代数完美导出范畴与某些C²上Lefschetz纤维的Fukaya-Seidel范畴等价，并给出了其Milnor纤维的完整描述。
These notes provide a description of the abelian categories that arise as categories of coherent sheaves on weighted projective lines. Two different approaches are presented: one is based on a list of axioms and the other yields a description in terms of expansions of abelian categories. A weighted projective line is obtained from a projective line by inserting finitely many weights. So we describe the category of coherent sheaves on a projective line in some detail, and the insertion of weights amounts to adding simple objects. We call this process `expansion' and treat it axiomatically. Thus…
We study the interaction between geometric operations on stacks and algebraic operations on their categories of sheaves. We work in the general setting of derived algebraic geometry: our basic objects are derived stacks X and their oo-categories QC(X) of quasicoherent sheaves. We show that for a broad class of derived stacks, called perfect stacks, algebraic and geometric operations on their categories of sheaves are compatible. We identify the category of sheaves on a fiber product with the tensor product of the categories of sheaves on the factors. We also identify the category of sheaves on…
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