Hua-Lin Huang, Shilin Yang
For a finite dimensional semisimple Lie algebra ${\frak{g}}$ and a root $q$ of unity in a field $k,$ we associate to these data a double quiver $\bar{\cal{Q}}.$ It is shown that a restricted version of the quantized enveloping algebras $U_q(\frak g)$ is a quotient of the double quiver algebra $k\bar{\cal Q}.$
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We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
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.
Inspired by the work of Rostam, we establish an explicit categorical equivalence between affine Yokonuma-Hecke algebras and quiver Hecke algebras associated to disjoint copies of quivers of (affine) type $A,$ generalizing Rouquier's categorical equivalence theorem.
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 introduce the notion of affinizations and R-matrices for arbitrary quiver Hekcke algebras. We show that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define the notion of a duality datum and construct a tensor functor between graded module categories of two quiver Hecke algebras. We give several examples of such functors .
In this paper, we compute the product structure of the Hochschild cohomology of preprojective algebras of quivers of type D and E over a field of characteristic zero.
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。
Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, $\mathfrak{n^{+}}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U(\mathfrak{n^{+}})$. In particular, the representation theory of tame quivers is essentially used in this paper.
Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.
We consider the Kronecker quiver and determine the relations for the specialisation to q=0 of the generic composition algebra as well as those for Reineke's composition monoid. We also obtain a normal form for the varieties occurring in the composition monoid in terms of Schur roots.
The Hochschild homology/cohomology an associative algebra, together with the Connes differential, the contraction map and the Lie derivative, forms the structure of calculus. In this paper we compute explicitely the calculus structure of preprojective algebras of Dynkin quivers over a field of characteristic zero. This also completes the work in math.AG/0502301, where the Batalin-Vilkovisky structure of the Hochschild cohomology of preprojective algebras of non-Dynkin quivers are computed and the calculus can be easily computed from that.
We construct bar-invariant $\mathbb{Z}[q^{\pm 1/2}]-$bases of the quantum cluster algebra of the Kronecker quiver which are quantum analogues of the canonical basis, semicanonical basis and dual semicanonical basis of the cluster algebra of the Kronecker quiver in the sense of \cite{sherzel},\cite{calzel} and \cite{gls} respectively. As a byproduct, we prove the positivity of the elements in these bases.
We construct all the irreducible representations of spin quiver Hecke algebras for orthosymplectic Lie superalgebras $osp(1|2n),$ and show that their highest weights are given by the dominant words. We use the dominant Lyndon words to construct the cuspidal modules and show that the irreducible representations are the simple heads of standard representations constructed by induction from the cuspidal modules.
Reflection functors on quiver Hecke algebras
本文针对任意可对称化 Kac-Moody 型的箭图 Hecke 代数构造反射函子，并证明其范畴化了 Lusztig 的辫子对称性。
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.
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.
We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of arXiv:1512.08533 and arXiv:1608.04651 with representation of fractional quivers.
We define elliptic generalization of W-algebras associated with arbitrary quiver using the formalism of arXiv:1512.08533 applied to six-dimensional quiver gauge theory compactified on elliptic curve.
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.
Double Quiver Gauge Theory and BPS/CFT Correspondence
本文提出基于 $q$-Cartan 矩阵的形式体系，用于计算多类流形上箭袋规范理论的瞬子配分函数，并引入八维双箭袋规范理论以探索 BPS/CFT 对应。
Quiver presentations and Schur--Weyl duality for Khovanov arc algebras
本文通过给出Khovanov弧代数的Ext-箭图与关系表示，证明了Kleshchev–Martin猜想在该框架下的精确类比，揭示了其与Schur–Weyl对偶的联系。
The main result of this paper shows that, over large enough fields of characteristic different from $2$, the alternating Hecke algebras are $\mathbb{Z}$-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group $\mathfrak{S}_n$. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from $2$, is a $\mathbb{Z}$-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras…
The quiver Yangian, an infinite-dimensional algebra introduced recently in arXiv:2003.08909, is the algebra underlying BPS state counting problems for toric Calabi-Yau three-folds. We introduce trigonometric and elliptic analogues of quiver Yangians, which we call toroidal quiver algebras and elliptic quiver algebras, respectively. We construct the representations of the shifted toroidal and elliptic algebras in terms of the statistical model of crystal melting. We also derive the algebras and their representations from equivariant localization of three-dimensional $\mathcal{N}=2$ supersymmetr…
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
Quiver moduli and quantum loop algebras
本文通过临界K理论将量子环形代数中所有简单模的q-特征与有向图模空间的欧拉特征联系起来。
Quantum loop groups for arbitrary quivers
本文研究任意箭图下量子环路群与Feigin-Odesskii型洗牌代数的对偶构造，统一涵盖K-理论Hall代数、Kac-Moody型量子环路群和量子环面代数等情形。
These notes reflect the contents of three lectures given at the workshop of the 14th International Conference on Representations of Algebras (ICRA XIV), held in August 2010 in Tokyo. We first provide an introduction to quantum loop algebras and their finite-dimensional representations. We explain in particular Nakajima's geometric description of the irreducible q-characters in terms of graded quiver varieties. We then present a recent attempt to understand the tensor structure of the category of finite- dimensional representations by means of cluster algebras.
Quiver Hecke algebras for Borcherds-Cartan datum III: Categorification of quantum Borcherds superalgebras
本文提出了一种有向图Hecke超代数，用于对量子Borcherds超代数进行分类化。
We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\mathcal{R}^Λ_n$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\mathcal{O}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of chara…
We apply a combinatorial formula of the first author and Rosso, for products in Hopf quiver algebras, to determine the structure of Nichols algebras. We illustrate this technique by explicitly constructing new examples of Nichols algebras in positive characteristic. We further describe the corresponding Radford biproducts and some liftings of these biproducts, which are new finite dimensional pointed Hopf algebras.
V. Ginzburg, and independently R. Bocklandt and L. Le Bruyn, defined an infinite-dimensional "necklace" Lie algebra canonically associated to any quiver. Following suggestions of V. Turaev, P. Etingof, and Ginzburg, we define a cobracket and prove that it defines a Lie bialgebra structure. We then present a Hopf algebra quantizing this Lie bialgebra, and prove that it is a Hopf algebra satisfying the PBW property. We present representations into spaces of differential operators on representations of the quiver, which quantize the trace representations of the Lie algebra given by Ginzburg.
We study finite dimensional representations of the quantum affine algebra, using geometry of quiver varieties introduced by the author. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0.
We provide a quiver setting for quasi-Hopf algebras, generalizing the Hopf quiver theory. As applications we obtain some general structure theorems, in particular the quasi-Hopf analogue of the Cartier theorem and the Cartier-Gabriel decomposition theorem.
Derivation based noncommutative geometry of quiver path algebras
本文研究箭图路代数的基于导子的非交换几何，引入分离箭图概念得到微分形式模为投射双模的等价条件，可构造Levi-Civita联络等非交换几何结构。
Categorification of quiver diagonalization and Koszul algebras
本文将对称箭图的连接与断开操作提升至二次代数层面，以此证明了每个对称箭图关联的超交换二次代数均满足Koszul性猜想。
Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
本文通过有向图方法构造了A型Dynkin图相关的Yang代数及其矩形表示，并展示了其与Gelfand-Tsetlin基的关联。
We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl_2), and to actions of the Drinfeld double of T(n).
Quiver connections and bimodules of basic algebras
本文为分类非半单有限维结合代数的量子对称性，定义了有界quiver间的connection并构建双范畴，证明其等价于basic algebras、bimodules与intertwiners的双范畴。
We consider the $k$-twisted Nekrasov-Shatashvili limit (NS$_k$ limit) of 5d (K-theoretic) and 6d (elliptic) quiver gauge theory, where one of the multiplicative equivariant parameters is taken to be the $k$-th root of unity. We obtain the extended center of the associated $q$-deformed quiver W-algebras constructed by our formalism [arXiv:1512.08533; arXiv:1608.04651; arXiv:1705.04410], which provides gauge theoretic proof of Bouwknegt-Pilch's statement on the relation to the representation ring of quantum affine algebra.
We survey some results on counting the rational points of moduli spaces of quiver representations. We then make generalizations to Grassmannians and flags of quiver representations. These results have nice applications to the cluster algebra. Along the way, we use the full Hopf structure of the Hall algebra of a quiver.
We describe certain quiver Hopf algebras by parameters. This leads to the classification of multiple Taft algebras as well as pointed Yetter-Drinfeld modules and their corresponding Nichols algebras. In particular, when the ground-field $k$ is the complex field and $G$ is a finite abelian group, we classify quiver Hopf algebras over $G$, multiple Taft algebras over $G$ and Nichols algebras in $^{kG}_{kG} {\cal YD}$. We show that the quantum enveloping algebra of a complex semisimple Lie algebra is a quotient of a semi-path Hopf algebra.
We study the Gauge/Bethe correspondence for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we…
Quiver algebras and their representations for arbitrary quivers
本文将箭图杨子代数从环面 Calabi–Yau 三维流形推广到任意带势箭图，并定义可涵盖动机 DT 不变量的精细化特征标。
We construct an $ε$-deformation of W algebras, corresponding to the additive version of quiver $\text{W}_{q,t^{-1}}$ algebras which feature prominently in the 5d version of the BPS/CFT correspondence and refined topological strings on toric Calabi-Yau's. This new type of algebras fill in the missing intermediate level between $q$-deformed and ordinary W algebras. We show that $ε$-deformed W algebras are spectral duals of conventional W algebras, in particular the $ε$-deformed conformal blocks manifestly reproduce instanton partition functions of 4d $\mathcal{N}=2$ quiver gauge theories in the …
Consider the Cohomological Hall Algebra as defined by Kontsevich and Soibelman, associated with a Dynkin quiver. We reinterpret the geometry behind the multiplication map in the COHA, and give an iterated residue formula for it. We show natural subalgebras whose product is the whole COHA (except in the $E_8$ case). The dimension count version of this statement is an identity for quantum dilogarithm series, first proved by Reineke. We also show that natural structure constants of the COHA are universal polynomials representing degeneracy loci, a.k.a. quiver polynomials.
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