Stefan Wolf
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.
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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 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.
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.
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。
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。
These are notes from my mini-course at ICRA13, Sao Paulo, 2008
We compute the Euler characteristics of quiver Grassmannians and quiver flag varieties of tree and band modules and prove their positivity. This generalizes some results by G.C. Irelli [arXiv:0910.2592]. As an application we consider the Ringel-Hall algebra $C(A)$ of some string algebras $A$ and compute in combinatorial terms the products of arbitrary functions in $C(A)$.
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.
We prove a Chevalley restriction theorem and its double analogue for the cyclic quiver.
We count the $\mathbb{F}_q$-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras -- one is one-point extended from a quiver $Q$, and the other is the Dynkin $A_2$ tensored with $Q$. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions.
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.
We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^σ$ where $\hat{\mathcal{H}}^σ$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^σ-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat…
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.
In this paper we describe the twisted Hall algebra of bound quiver with small homological dimension. The description is given in the terms of the quadratic form associated with the corresponding bound quiver.
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 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…
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.
In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.
Maximal Rigid Representations of Continuous Quivers of Type A with Automorphism
本文利用Buan和Krause对循环箭图最大刚性表示的分类结果，推导出了带自同构的A型连续箭图一类最大刚性表示的同构类数量公式。
We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov, and we construct an embedding of Y in the Yangian, intertwining the respective actions of both algebras on the cohomology of quiver varieties.
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 …
Cohomological Hall algebras of quivers and Yangians
该研究建立了任意箭图的预投射上同调霍尔代数与对应Maulik-Okounkov Yangian正半部分的同构关系，且该同构保持二者在中岛箭图簇上同调的作用，进而证明了Okounkov关于李代数特征与Kac多项式关联的猜想。
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 derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver.
We compute two representations of COHA for $A_1$-quiver. The two untwisted representations can be combined into a representation of $D_{n+1}$ Lie algebra. The untwisted increasing representation and the twisted decreasing representation can be combined into a representation of a finite Clifford algebra.
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.
Associated graded modules of mixed Hodge modules on quivers and preprojective K-theoretic Hall algebra
通过取相伴分次模研究箭图模空间上混合霍奇模与预投射代数的 K - 理论霍尔代数间的联系，还证明了取相伴分次模的函子与反射函子可交换。
In this paper we give a geometric construction of the quantum group Ut(G) using Nakajima categories, which were developed in [29]. Our methods allow us to establish a direct connection between the algebraic realization of the quantum group as Hall algebra by Bridgeland [1] and its geometric counterpart by Qin [24].
Hall Geometry and Auslander-Reiten Quiver
本文利用动机Hall代数中的几何信息和模堆的对应关系，恢复Auslander-Reiten序列和Auslander-Reiten箭图。
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…
We use the comultiplication to prove that Hall polynomials exist for all finite and affine quivers. In the finite and cyclic cases, this approach provides a new and simple proof of the existence of Hall polynomials. In general, these polynomials are defined with respect to the decomposition classes of Bongartz and Dudek, a generalisation of the Segre classes for square matrices.
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。
Geometric realizations of Ringel-Hall algebras of continuous quivers of type $A$
本文通过卢斯蒂格的几何实现方法，利用近似方法，实现了类型A连续四元组的环形霍尔代数。
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_…
We give a short introduction to Cohomological Hall algebras of quivers and describe the semistable Cohomological Hall algebra of central slope of the Kronecker quiver in terms of generators and relations.
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。
For $(Q,W)$ a symmetric quiver with potential satisfying a Künneth-type condition, we construct (positive and negative) extensions of its K-theoretic Hall algebra which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs of a quiver $Q$ and one can construct their Drinfeld double algebra.
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。
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…
We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac co…
By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in $\mathbf K$ form the canonical basis of $\mathbf K$. The relationships among the algebra $\mathbf K$, the composition algebra of locally projective representations of quivers over $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided.
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 is an overview article on finite-dimensional algebras and quivers, written for the Encyclopedia of Mathematical Physics. We cover path algebras, Ringel-Hall algebras and the quiver varieties of Lusztig and Nakajima.
We establish a connection between a generalization of KLR algebras, called quiver Schur algebras, and the cohomological Hall algebras of Kontsevich and Soibelman. More specifically, we realize quiver Schur algebras as algebras of multiplication and comultiplication operators on the CoHA, and reinterpret the shuffle description of the CoHA in terms of Demazure operators. We introduce ``mixed quiver Schur algebras" associated to quivers with a contravariant involution, and show that they are related, in an analogous way, to the cohomological Hall modules defined by Young. We also obtain a geomet…
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。
The semi-stable ChowHa of a quiver with stability is defined as an analog of the Cohomological Hall algebra of Kontsevich and Soibelman via convolution in equivariant Chow groups of semi-stable loci in representation varieties of quivers. We prove several structural results on the semi-stable ChowHa, namely isomorphism of the cycle map, a tensor product decomposition, and a tautological presentation. For symmetric quivers, this leads to an identification of their quantized Donaldson-Thomas invariants with the Chow-Betti numbers of moduli spaces.
For Dynkin quivers, we find the Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ and use $\widetilde{X}_{a, c}^{b}(v)$ to construct the Hall algebra $\hc_v(\cc(\cp))$ over $\mz[v, v^{-1}]$, where $\widetilde{X}_{a, c}^{b}(|\mf_q|)$'s are structure constants used by Bridgeland. The Laurent polynomials $\widetilde{X}_{a, c}^{b}(v)$ are explicitly given in $A_1$ case. As an application, we obtain the full quantum groups $U_t(\sg)$ associated to the Dynkin quivers for arbitrary $t\not=0,\pm1$.
The $\imath$quiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal $\imath$quantum groups, which is a generalization of Bridgeland's Hall algebra construction for (Drinfeld doubles of) quantum groups; here an $\imath$quantum group and a corresponding Drinfeld-Jimbo quantum group form a quantum symmetric pair. In this paper, the Dynkin $\imath$quiver algebras are shown to arise as new examples of singular Nakajima-Keller-Scherotzke categories. Then we provide a geometric construction of the universal $\imath$quantum groups and their ``dual can…
C.M. Ringel defined Hall algebra associated with the category of representations of a quiver of Dynkin type and gave an explicit description of the structure constants of the corresponding Lie algebra. We utilize functorial properties of the Hall algebra to give a simple proof of Ringel's result, and to generalize it to the case of a quiver of affine type.
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