Tim Cramer
We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of [BS1], [BS2], [SVdB], and [XY].
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Double Hall algebras and derived equivalences revisited
本文针对Cramer关于双Hall代数与导出等价的定理，修正了其证明中依赖的特征三角计数公式错误，给出保留原策略的新证明。
Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.
In [19], Zheng studied the bounded derived categories of constructible $\bar{\mathbb{Q}}_l$-sheaves on some algebraic stacks consisting of the representations of a enlarged quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we shall generalize Zheng's work to highest weight modules of a subalgebra of the double Ringel-Hall algebra associated to a quiver in a functional version.
The elliptic Hall algebra and the double Dyck path algebra
本研究证明椭圆霍尔代数正半部分可嵌入双戴克路径代数作为球面子代数，还定义了双戴克路径代数的双重版本以构造完整椭圆霍尔代数。
In this paper, we study the category $\mathscr{H}^{(ρ)}$ of semi-stable coherent sheaves of a fixed slope $ρ$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(ρ)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.
Let $\mathcal{A}$ be an arbitrary hereditary abelian category that may not have enough projective objects. For example, $\mathcal{A}$ can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of $\mathcal{A}$, denoted by $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathcal{A})$, to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of $\mathbb{Z}/2$-gr…
Group Cohomology and Gauge Equivalence of some Twisted Quantum Doubles
Dijkgraaf，Pasquier和Roche在共形域论的背景下引入了有限群的扭曲量子双打。我们研究了与这些量子双打相关联的辫状幺半范畴范畴之间的等价性，特别是在交换的情况下。这涉及到一个密切的研究上同调的各种复杂的介绍艾伦伯格-麦克莱恩。我们证明了编织幺半范畴范畴的等价性与相应的量子偶的规范等价性是相同的，并且等价类的不变量由代谢二次空间给出。
Double Hall-Littlewood symmetric polynomials
本文通过建立环同构，将导出霍尔代数与双重对称函数环联系起来，揭示了双重霍尔-利特伍德对称函数的结构与性质，并推导了相关规则和生成函数。
We investigate the structure of the double Ringel-Hall algebras associated with cyclic quivers and its connections with quantum loop algebras of $\mathfrak{gl}_n$, affine quantum Schur algebras and affine Hecke algebras. This includes their Drinfeld-Jimbo type presentation, affine quantum Schur-Weyl reciprocity, representations of affine quantum Schur algebras, and connections with various existing works by Lusztig, Varagnolo-Vasserot, Schiffmann, Hubery, Chari-Pressley, Frenkel-Mukhin, etc. We will also discuss conjectures on a realization of Beilinson-Lusztig-MacPherson type and Lusztig type…
Let $A$ be a finite dimensional hereditary algebra over a finite field, and let $m$ be a fixed integer such that $m=0$ or $m>2$. In the present paper, we first define an algebra $L_m(A)$ associated to $A$, called the $m$-periodic lattice algebra of $A$, and then prove that it is isomorphic to Bridgeland's Hall algebra $\mathcal {D}\mathcal {H}_m(A)$ of $m$-cyclic complexes over projective $A$-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of $A$ into $\mathcal {D}\mathcal {H}_m(A)$.
Let $\A$ be a finitary hereditary abelian category and $D(\A)$ be its reduced Drinfeld double Hall algebra. By giving explicit formulas in $D(\A)$ for left and right mutations, we show that the subalgebras of $D(\A)$ generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if $\A$ is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of $\A$ is generated by any complete exceptional sequence. Moreover, for …
For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.
We show that the Hall algebra of two-periodic complexes, which is recently introduced by T.Bridgeland, coincides with the Drinfeld double of the ordinary Hall bialgebra.
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups
我们建立了有限群的扭曲量子对偶上的模范畴之间的辫张量等价，该有限群是由群H的一个阿贝尔群扩张而成的，H上的一个3-上圈是3-上圈的膨胀。我们还证明了有限群的任意扭曲量子偶模范畴的正则带结构由辫状张量等价保持。我们给出两个主要应用：首先，如果G是宽度至少为2的特殊2-群，我们证明了G的被3-上圈w扭曲的量子偶与初等阿贝尔2-群的扭曲量子偶规范等价当且仅当w^2是平凡的;其次，我们讨论了8阶群的扭曲量子偶的规范等价类，并对这些拟三角拟双代数的辫子张量等价类进行了分类。事实证明，正好有20个这样的等价类。
Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
对有限群的扭曲量子偶的表示范畴进行了拉格朗日子范畴的分类。鉴于0704.0195v2的结果，这给出了有限群的扭曲量子偶的所有编织张量等价对的完整描述。我们还建立了有限群G的扭曲量子对偶的表示范畴的拉格朗日子范畴与扭曲G-分次向量空间范畴上的模范畴之间的典范双射，使得对偶张量范畴是指向的.这可以被看作是V. Drinfeld的量子版本的齐性空间的泊松-李群的特征在于拉格朗日子代数的双重李代数。由此得到两个群论融合范畴弱Morita等价的充要条件是它们的中心等价于辫状张量范畴。
On Gauge Equivalence of Twisted Quantum Doubles
本文研究了3-上圈扭曲的有限阿贝尔群量子双的规范等价条件，并应用于余根分次有限维余拟Hopf代数的分类及Nichols代数的无限维性证明。
We study the bialgebra structure of the Hall algebra of two-periodic complexes recently introduced by Bridgeland. We introduce coproduct on Bridgeland's Hall algebra, and show that in the hereditary case the resulting bialgebra structure coincides with that on Drinfeld double of the ordinary Hall algebra
The Hopf algebra $\mathcal{D}$ which is the subject of this paper can be viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime and primitive spectra are completely determined. As a corollary of this analysis it is shown that $\mathcal{D}$ satisfies the Dixmier-Moeglin Equivalence, leading to the formulation of a conjecture on the validity of this equivalence for pointed Noetherian Hopf algebras.
We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the composition subalgebra for a smooth elliptic curve. As in the case of smooth elliptic case studied by Burban and Schiffmann, the reduced Drinfeld double of the composition subalgebra is isomorphic to the quantum toroidal algebra for $\mathfrak{gl}_1$ (also called Ding-Iohara-Miki algebra), and it inherits automorphisms induced from equivalences of the associated de…
$K$-theoretic Hall algebras and Coulomb branches
本文通过洗牌代数解释，构建了一个从合适解释的双重环-nilpotent $K$-理论霍尔代数到quiver规范理论色姆分支代数的满同态，解决了霍尔代数与色姆分支之间的联系问题。
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