Love Forsberg
We describe the combinatorics of the multisemigroup with multiplicities for the tensor category of subbimodules of the identity bimodule, for an arbitrary non-uniform orientation of a finite cyclic quiver.
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Harish-Chandra bimodules over quantized flower quiver varieties with minimal support
本研究对具有极小支撑的量子化花箭簇簇上的Harish-Chandra双模进行分类，证明维数向量为$n$时，仅当整数量化参数足够大或足够小时存在$n!$个极小支撑单Harish-Chandra双模，核心工具为增强版限制函子。
We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical square zero relations. For such algebras we study the (finitary) tensor category of bimodules. We describe the cell structure of this tensor category, determine existing adjunctions between its $1$-morphisms and find a minimal generating set with respect to the tensor structure. We also prove that, for the algebras mentioned above, every simple transitive $2$-r…
The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver $Q$, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that w…
We construct an explicit projective bimodule resolution for the Leavitt path algebra of a row-finite quiver. We prove that the Leavitt path algebra of a row-countable quiver has Hochschild cohomolgical dimension at most one, that is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the resolution relies on an explicit derivation of the Leavitt path algebra.
We introduce and study potentials, mutations and Jacobian algebras in the framework of tensor algebras associated with symmetrizable dualizing pairs of bimodules on a symmetric algebra over any commutative ground ring. The graded context is also considered by starting from graded bimodules, and the classical non simply-laced context of modulated quivers with potentials is a particular case. The study of potentials in this framework is related to symmetrically separable algebras, and we have two kinds of potentials: the symmetric and the non symmetric ones. When the Casimir ideal of the symmetr…
From brick manifolds to Grassmannians of bimodules
本文研究箭图路径代数上子双模的格拉斯曼流形，推广了砖流形，给出显式构造、光滑性证明、胞腔分解及带框模空间实现，并导出动机的递归公式。
We study quantum dilogarithm identities for cyclic quivers following Reineke's idea via Ringel-Hall algebra approach. For any given discrete stability function for the cyclic quiver $Δ_n$ with $n$ vertices, we obtain certain cyclic quantum dilogarithm identities of order $n$ in the sense of Bytsko and Volkov.
In the case of cyclic quiver we prove that the deformed Harish-Chandra map whose existence was conjectured by Etingof and Ginzburg is well defined. As an application we prove Kirillov-type formula for the cyclotomic Bessel function.
Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic. We call $Q$ cluster-acyclic in this case, otherwise cluster-cyclic. In this note we classify the cluster-cyclic quivers with three vertices using a Diophantine equation studied by Markov.
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
Cyclically ordered quivers
本文研究了具有顶点循环顺序结构的有向图，这种结构在许多重要应用中自然出现，并引入了新的强大突变不变量。
We show that the spherical subalgebra of the rational Cherednik algebra associated to the wreath product of a symmetric group and a cyclic group is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver. This confirms a version of a conjecture of Etingof and Ginzburg in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov on the deformed Harish-Chandra homomorphism, and of Crawley-Boevey and of Gan and Ginzburg on preprojective algebras.
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster category of finite dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character (in the sense of \cite{Palu}) and satisfies some inductive formulas for the multiplication between the generalized cluster variables (the images of objects of the cluster category under the map). Moreover, we construct a $\mathbb{Z}$-basis for the algebras generated by all generalized cluster variables.
For a Dynkin quiver $Q$ of type ADE and a sum $β$ of simple roots, we construct a bimodule over the quantum loop algebra and the quiver Hecke algebra of the corresponding type via equivariant K-theory, imitating Ginzburg-Reshetikhin-Vasserot's geometric realization of the quantum affine Schur-Weyl duality. Our construction is based on Hernandez-Leclerc's isomorphism between a certain graded quiver variety and the space of representations of the quiver $Q$ of dimension vector $β$. We identify the functor induced from our bimodule with Kang-Kashiwara-Kim's generalized quantum affine Schur-Weyl d…
We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras $A$ and $B$, we use the special monomorphism category Mon(B, A-Gproj) to describe some Gorenstein projective bimodules over the tensor product of $A$ and $B$. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for Mon(B, A-Gproj) being the category of all Gorenstein projective bimodules. In addition, If both $A$ and $B$ are Gorenstein, we can describe the category of all Gorens…
These are notes from my mini-course at ICRA13, Sao Paulo, 2008
We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver.
Primitive elements in the Hall algebra of a cyclic quiver
本文研究循环箭图幂零表示的Hall代数，给出了其中本原元素的显式公式，解决了该类代数中本原元的具体构造问题。
We obtain an explicit combinatorial formula for certain parabolic Kostka-Shoji polynomials associated with the cyclic quiver, generalizing results of Shoji and of Liu and Shoji.
Finite dimensional 2-cyclic Jacobian algebras
本文证明仅含2-环和环的箭图上存在Jacobian-有限势，并利用覆盖理论与2-循环Caldero-Chapoton公式，在三种特定曲面情形下用τ-刚性模范畴化Paquette-Schiffler广义丛代数。
We define a set of "enhanced" nilpotent quiver representations that generalizes the enhanced nilpotent cone. This set admits an action by an associated algebraic group $K$ with finitely many orbits. We define a combinatorial set that parametrizes the set of orbits under this action and we derive a purely combinatorial formula for the dimension of an orbit.
In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver.Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same…
We prove that the local intersection cohomology of nilpotent orbit closures of cyclic quivers is trivial when the two orbits involved correspond to partitions with at most two rows. This gives a geometric proof of a result of Graham and Lehrer, which states that standard modules of the affine Hecke algebra of $GL_d$ corresponding to nilpotents with at most two Jordan blocks are multiplicity-free.
We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $\tt A$ flag varieties, their linear degenerations, finite dimensional approximations of the $GL_n$-affine flag variety and affine Grassmannian. We show that these quiver Grassmannians, equipped with our torus action, are GKM-varieties and that their moment graph admits a combinatorial description in terms of coefficients quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able …
The intersection cohomologies of closures of nilpotent orbits of linear (respectively, cyclic) quivers are known to be described by Kazhdan-Lusztig polynomials for the symmetric group (respectively, the affine symmetric group). We explain how to simplify this description using a combinatorial cancellation procedure, and derive some consequences for representation theory.
We calculate the additive and multiplicative structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types T. We also compute the calculus structure which is formed by the Hochschild homology/cohomology pair.
We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the numb…
If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example tha…
We calculate the additive structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types ADE.
The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups
本文研究具有循环亏群的块的完美p-置换双模表示环的环论结构，证明嘉当矩阵含初等因子1时[B]非本原，并给出本原分解及标量扩张后的直积分解。
In this paper, we give a recursive formula for the interesting PBW basis $E_{A}$ of composition subalgebras of Ringel-Hall algebras $\fkH_\vartri(n)$ of cyclic quivers after \cite{DengDuXiao2007generic}, and another construction of canonical bases of $\U_v^+(\wih{\fks\fkl}_n)$ from the monomial bases $m^{(A)}$ follow \cite{DuZhaomultiplication}. As an application, we will determined all the canonical basis of $\U_v^+(\wih{\fks\fkl}_2)$ associated with modules of Lowery length $\leqs3$. Finally, we will discuss the relation of canonical bases of Ringel-Hall algebras and those of affine quantum …
This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras $E$ over a field $F$, such that $E \otimes_{F} E^{op}$ is semisimple. We assume that $E$ contains a certain type of $F$-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over $E$, of any finite-dimensional $E$-$E$-bimodule on which $F$ acts centrally. In this case, we introduce a cyclic derivative and to each potential we associ…
Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.
Quiver Grassmannians associated to nilpotent cyclic representations defined by single matrix
本文研究由单个矩阵定义的幂零循环表示所关联的箭图Grassmannian的闭Białynicki-Birula胞腔几何，证明在子表示维数为全1时这些胞腔光滑，并借助Knutson-Tao基描述了其上同调环的乘法结构。
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig's bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland's recent work for type ADE.
Nilpotent representations over equioriented cyclic quivers
该研究通过显式双射刻画等定向循环箭图的幂零表示，在有限域上推导相关计数与概率公式，在布尔半环上关联幂零半表示与有向无环图并确定概率渐近衰减率。
The $\mathrm{GL}(V)$-orbits in the enhanced nilpotent cone $V\times\mathcal{N}(V)$ are (essentially) in bijection with the orbits of a certain parabolic $P\subseteq\mathrm{GL}(V)$ (the mirabolic subgroup) in the nilpotent cone $\mathcal{N}(V)$. We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different to the previous ones that have appeared in the literature. Explicit translations between the different pa…
Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel--Hall algebra ${\mathscr D}(Q)$ of a quiver $Q$ as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation $L(Λ_0)$ of ${\mathscr D}(Δ_n)$ of the cyclic quiver $Δ_n$ provides a realization of the $q$-deformed Fock space $\bigwedge^\infty$ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of $Δ_n$, we obtain a decomposition of the basic representation $L(Λ_…
Dimer algebras, ghor algebras, and cyclic contractions
本文研究环面上 ghor 代数与二聚体代数的关系，证明诺特性等价于二聚体代数，并分类最大维数简单模及描述中心，引入 Higgsing 和介子手性环形式化概念。
The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The s…
Symplectic Grassmannians and Cyclic Quivers
本文将箭图Grassmann流形方法推广到辛Grassmann簇退化情形，构造胞腔分解、计算Euler示性数与Poincaré多项式，并证明不可约分支数等于经典辛Grassmann流形的Euler示性数。
Normed representations of weight quivers
本文为由权箭图给出的张量环和双模引入范数，定义了以三元组为对象的范畴 $\mathscr{A}^p_ς$，并证明其存在初始对象，使得多种经典积分与级数展开均可视为该范畴中的态射。
Harish-Chandra bimodules for type A rational Cherednik algebras
本文研究A型有理Cherednik代数的Harish-Chandra双模，证明其范畴可完全忠实嵌入范畴O_c，并在Nakajima箭图簇量子化的一般框架下构造对偶，进而描述含正则双模的最小Serre子范畴及其不可约对象的张量积。
Stable Graded Multiplicities for Harmonics on a Cyclic Quiver
本文研究循环箭图对应的 Vinberg θ-群，将调和函数视为一般线性群乘积 K 的表示，通过组合分离变量消去不变式，给出了每个 K-型稳定分次重数的组合公式。
This short note contains a combinatorial construction of symmetries arising in symplectic geometry (partially wrapped or infinitesimal Fukaya categories), algebraic geometry (derived categories of singularities), and K-theory (Waldhausen's S-construction). Our specific motivation (in the spirit of expectations of Kontsevich, and to be taken up in general elsewhere) is a combinatorial construction of quantizations of Lagrangian skeleta (equivalent to microlocal sheaves in their many guises). We explain here the one dimensional case of ribbon graphs where the main result of this paper gives an i…
CoHA of Cyclic Quivers and an Integral Form of Affine Yangians
本文研究了循环有向图的CoHA，揭示其与有理Yangian的积分形式的关系，并探讨了相关代数结构与猜想。
Hopf actions of some quantum groups on path algebras
本文研究量子Borel子代数和小量子群在箭图路径代数上的Hopf作用，给出参数化及单位根情形下作用分解的充要条件，并借助张量范畴语言将相应双模范畴等价于明确关系箭图的表示范畴。
Module categories, internal bimodules and Tambara modules
本文利用Tambara模理论将刚性幺半范畴上模范畴的重构定理推广至非刚性情形，证明了循环模范畴与Tambara模代数双模对象的双等价性，并给出了“通过丰富化实现作用”的嵌入结果。
We prove that the KLR algebra associated with the cyclic quiver of length $e$ is a subquotient of the KLR algebra associated with the cyclic quiver of length $e+1$. We also give a geometric interpretation of this fact. This result has an important application in the theory of categorical representations. We prove that a category with an action of $\widetilde{\mathfrak{sl}}_{e+1}$ contains a subcategory with an action of $\widetilde{\mathfrak{sl}}_{e}$. We also give generalizations of these results to more general quivers and Lie types.
We investigate the combinatorics of quivers that arise from triangulations of even-dimensional cyclic polytopes. Work of Oppermann and Thomas pinpoints such quivers as the prototypes for higher-dimensional cluster theory. We first show that a $2d$-dimensional triangulation has no interior $(d + 1)$-simplices if and only if its quiver is a cut quiver of type $A$, in the sense of Iyama and Oppermann. This is a higher-dimensional generalisation of the fact that triangulations of polygons with no interior triangles correspond to orientations of an $A_{n}$ Dynkin diagram. An application of this fir…
On higher extensions of quiver representations over $\mathbb{F}_1$
本文通过循环箭图的例子，证明了有限维幂零 $\mathbb{F}_1$-表示之间的高次扩张空间可能是无限维的，澄清了文献中的一个误解，并指出对于循环箭图 $\Delta_n$，$\operatorname{Ext}^3(-,-)$ 消失而 $\operatorname{Ext}^2(-,-)$ 对简单表示是无限维的。
We study the Hall and composition algebras of an affine quiver. In the case of a cyclic quiver, we provide generators for the central polynomial algebra described by Schiffmann and prove that this is in fact the whole of the centre of the Hall algebra. For an affine quiver without oriented cycles, we obtain a structure theorem for the composition algebra refining the structure provided by Zhang.
The monomorphism category $\mathscr{S}(A, M, B)$ induced by a bimodule $_AM_B$ is the subcategory of $Λ$-mod consisting of $\left[\begin{smallmatrix} X\\ Y\end{smallmatrix}\right]_ϕ$ such that $ϕ: M\otimes_B Y\rightarrow X$ is a monic $A$-map, where $Λ=\left[\begin{smallmatrix} A&M\\0&B \end{smallmatrix}\right]$. In general, it is not the monomorphism categories induced by quivers. It could describe the Gorenstein-projective $\m$-modules. This monomorphism category is a resolving subcategory of $\modcatΛ$ if and only if $M_B$ is projective. In this case, it has enough injective objects and Aus…
Faithfulness of the 2-Braid Group via Zigzag Algebra in Type B
本文通过证明B型之字形代数上的双模范畴是B型Soergel双模范畴的商范畴，为Rouquier关于B型2-辫子群忠实性的猜想提供了另一种证明。
Wreath Macdonald polynomials, quiver varieties, and quasimap counts
本研究通过wreath Macdonald多项式与量子环面代数，推导了循环Nakajima箭图簇上带帽顶点函数的显式生成公式，并加强了经典性结果，提出积分性与墙交叉猜想。
Calabi-Yau completions for roots of dualizing dg bimodules
本文针对 dg 范畴上移位逆对偶化双模的根，引入了 $a$ 次根对及其 Calabi-Yau 完备化，证明了其循环不变性条件，并建立了与 Adams 分次 Calabi-Yau dg 范畴的双射，进而推广了簇范畴及倾斜丛定理。
We define and study the category $\RepQ$ of representations of a quiver in $\VFun$ - the category of vector spaces "over $\Fun$". $\RepQ$ is an $\Fun$-linear category possessing kernels, co-kernels, and direct sums. Moreover, $\RepQ$ satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra $\HQ$ of $\RepQ$, which behaves in some ways like the specialization at $q=1$ of the Hall algebra of $\on{Rep}(\Q, \mathbf{F}_q)$. We prove the existence of a Hopf algebra homomorphism of $ ρ': \U(\n_+) \rightarrow \HQ$, from the enveloping algebra of t…
We study rotation of invariant vectors in tensor products of minuscule representations. We define a combinatorial notion of rotation of minuscule Littelmann paths. Using affine Grassmannians, we show that this rotation action is realized geometrically as rotation of components of the Satake fibre. As a consequence, we have a basis for invariant spaces which is permuted by rotation (up to global sign). Finally, we diagonalize the rotation operator by showing that its eigenspaces are given by intersection homology of quiver varieties. As a consequence, we generalize Rhoades' work on the cyclic s…
Let $A$ be a Nakayama algebra. We give a description of the singularity category of $A$ inside its stable module category. We prove that there is a duality between the singularity category of $A$ and the singularity category of its opposite algebra. As a consequence, the resolution quiver of $A$ and the resolution quiver of its opposite algebra have the same number of cycles and the same number of cyclic vertices.
In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the $q$-ho…
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