Shintarou Yanagida
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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
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.
Let $A$ be a finite dimensional algebra of finite global dimension over a finite field. In the present paper, we introduce certain elements in Bridgeland's Hall algebra of $A$, and give a multiplication theorem of these elements. In particular, this generalizes the main result in $[4]$.
Inspired by recent work of Bridgeland, from the category C^b(E) of bounded complexes over an exact category E satisfying certain finiteness conditions, we construct an associative unital "semi-derived Hall algebra" SDH(E). This algebra is an object sitting, in some sense, between the usual Hall algebra H(C^b(E)) and the Hall algebra of the bounded derived category D^b(E), introduced by Toen and further generalized by Xiao and Xu. It has the structure of a free module over a suitably defined quantum torus of acyclic complexes, with a basis given by the isomorphism classes of objects in the boun…
In this note, let $\A$ be a finitary hereditary abelian category with enough projectives. By using the associativity formula of Hall algebras, we give a new and simple proof of the main theorem in \cite{Yan}, which states that the Bridgeland's Hall algebra of 2-cyclic complexes of projective objects in $\A$ is isomorphic to the Drinfeld double Hall algebra of $\A$. In a similar way, we give a simplification of the key step in the proof of Theorem 4.11 in \cite{LP}.
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 $\mathcal{A}$ 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…
We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.
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 $t\not=0,\pm1$.
Let $\A$ be a finitary hereditary abelian category with enough projectives. We study the Hall algebra of complexes of fixed size over projectives. Explicitly, we first give a relation between Hall algebras of complexes of fixed size and cyclic complexes. Secondly, we characterize the Hall algebra of complexes of fixed size by generators and relations, and relate it to the derived Hall algebra of $\A$. Finally, we give the integration map on the Hall algebra of $2$-term complexes over projectives.
We establish an embedding of the quantum enveloping algebra of a symmetric generalized Kac--Moody algebra into a localized Hall algebra of $\mathbb Z_2$-graded complexes of representations of a quiver with (possible) loops. To overcome difficulties resulting from the existence of infinite dimensional projective objects, we consider the category of finitely-presented representations and the category of $\mathbb Z_2$-graded complexes of projectives with finite homology.
Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ be the category of 1-cyclic complexes of projective $A$-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra $\H(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in $\H(C_1(\mathscr{P}))$, and obtain quantum Serre relations…
$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type
本文通过引入带对合范畴上的 $\varrho$-复形并建立其同调性质，构造加权射影直线的 $\imath$Hall 代数，进而实现拟分裂仿射 ADE 型 $\imath$ 量子环代数。
Periodic derived Hall algebras of hereditary abelian categories
本文利用有界导出范畴的导出Hall数，为遗传阿贝尔范畴的周期导出范畴定义m-周期扩张导出Hall代数，从而统一显式刻画Bridgeland周期复形Hall代数及Xu-Chen奇周期导出Hall代数的代数结构。
$\imath$Hall algebras and $\imath$quantum groups
本文探讨了$\imath$Hall代数的构造及其在$\imath$量子群中的应用，包括通过反射functor实现对称性和与对称函数的联系。
Quantum symmetric pairs via Hall algebras
本文通过霍尔代数构造了量子对称对的结构，并证明了相关映射保持积分形式，用于构建对偶 canonical 基。
Given a Frobenius category $\mathcal{F}$ satisfying certain finiteness conditions, we consider the localization of its Hall algebra $\mathcal{H(F)}$ at the classes of all projective-injective objects. We call it the {\it "semi-derived Hall algebra"} $\mathcal{SDH(F, P(F))}.$ We discuss its functoriality properties and show that it is a free module over a twisted group algebra of the Grothendieck group $K_0(\mathcal{P(F)})$ of the full subcategory of projective-injective objects, with a basis parametrized by the isomorphism classes of objects in the stable category $\underline{\mathcal{F}}$. We…
Kleinian orbifolds, Cohomological Hall Algebras, and Yangians
本文为亏格解析克莱因奇点的轨形建立支撑例外轨迹凝聚层的COHA存在性并计算为仿射杨代数正半部分完备化，证明布里奇兰德稳定性条件空间点源自克莱因轨形且仿射杨代数正半部分可从对应COHA恢复，提供首个稳定性条件空间上的点态COHA族例子。
A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbolς}$ (called an $\imath$quantum group) with parameters ${\boldsymbolς}$. In this note, we use the derived Hall algebras of 1-periodic complexes to realize the $\imath$quantum groups ${\mathbf U}^{\imath}_{\boldsymbolς}$ of split type.
Lie algebras arising from two-periodic projective complex and derived categories
本文利用二周期射影复形范畴及其稳定范畴，分别为有限维代数的模范畴构造了李代数，并证明了这两种构造在自然函子与 Grothendieck 群同构下诱导出李代数同构，从而统一了两种结构常数的计算。
$\imath$Hall algebra of the projective line and $q$-Onsager algebra
本文证明射影直线的 $\imath$Hall 代数在 Drinfeld 型呈现下实现泛 $q$-Onsager 代数，并通过导出等价建立其与 Kronecker 箭图 $\imath$Hall 代数的同构，统一了两种呈现。
We give a geometric formulation of Toën's derived Hall algebra by constructing Grothendieck's six operations for the derived category of lisse-étale constructible sheaves on the derived stacks of complexes. Our formulation is based on an variant of Laszlo and Olsson's theory of derived categories and six operations for algebraic stacks. We also give an $\infty$-theoretic explanation of the theory of derived stacks, which was originally constructed by Toën and Vezzosi in terms of model theoretical language.
$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs
本文通过加权射影直线的 $\imath$Hall 代数实现 $\imath$ 量子环路代数，并利用导出等价建立两种展示下 $\imath$ 量子群的同构。
Cohomological Hall algebras, their categorification, and their representations via torsion pairs
本文系统构造上同调、K-理论和范畴化 Hall 代数的表示，统一恢复并范畴化箭图 COHA 及层 COHA 的已知作用，并通过挠对给出新的正负算子作用，同时推广 Khan 的 motivic Borel-Moore 同调以产生新 COHA 例子。
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simp…
Let A be an abelian category of finite type and homological dimension 1. Then by results of Green R(A), the extended Hall-Ringel algebra of A, has a natural Hopf algebra structure. We consider its Heisenberg double Heis(A) and study its relation with D(A), the derived category of A. We show that Heis(A) can be viewed as a "Hall algebra" of D^{0,1}(A), the subcategory of complexes situated in degrees 0 and 1, in the following sense: if B is the heart of a t-structure on D(A) lying in D^{0,1}(A), then R(B) is naturally a subalgebra in Heis(A). Further, we define a new algebra L(A) called the lat…
Perverse coherent extensions on Calabi-Yau threefolds and representations of cohomological Hall algebras
本文在环面 Calabi–Yau 三维流形上定义并研究带标架的反常凝聚扩张栈，将其与带势箭图、上同调 Hall 代数、DT/PT 理论及 AGT 相关表示联系起来。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。