Ming Lu, Weiqiang Wang
We establish a Drinfeld type new presentation for the $\imath$quantum groups arising from quantum symmetric pairs of split affine ADE type, which includes the $q$-Onsager algebra as the rank 1 case. This presentation takes a form which can be viewed as a deformation of half an affine quantum group in the Drinfeld presentation.
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Recently, Lu and Wang formulated a Drinfeld type presentation for $\imath$quantum group $\widetilde{\mathbf U}^\imath$ arising from quantum symmetric pairs of split affine ADE type. In this paper, we generalize their results by establishing a current presentation for $\widetilde{\mathbf U}^\imath$ of arbitrary split affine type.
$\imath$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。
Let $(\bf U, \bf U^\imath)$ be a quantum symmetric pair of Kac-Moody type. The $\imath$quantum groups $\bf U^\imath$ and the universal $\imath$quantum groups $\widetilde{\bf U}^\imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $\widetilde{\bf U}$. In this paper we formulate and establish Serre-Lusztig relations for $\imath$quantum groups in terms of $\imath$divided powers, which are an $\imath$-analog of Lusztig's higher order Serre relations for quantum groups. This has applications to braid group symmetries on $\imath$quantum groups.
Braid group action and quasi-split affine $\imath$quantum groups I
本文聚焦实秩一拟分裂仿射量子对称对，显式构造了 $A_2^{(2)}$ 型相对辫群作用，建立了实虚根向量与 Drinfeld 型展示，为后续高秩情形奠定基础。
Affine $\imath$quantum groups and twisted Yangians in Drinfeld presentations
本文通过仿射 ı 量子群 Drinfeld 呈现的退化构造了分裂型扭 Yangian，证明其具有 PBW 型基且为扭流代数的形变，并猜想其与 RTT 呈现中的对应代数同构。
Braid group action and quasi-split affine iquantum groups III
本文是系列研究第三篇，通过构造实与虚$v$-根向量并建立其关系，为剩余的${\rm AIII}^{(τ)}_{2r}$型拟分裂仿射i量子群提供了Drinfeld表示。
Analogue of Feigin's map on $\imath$quantum group of split type
针对分裂型泛$\boldsymbol{\tau}$量子群这一量子群的广泛推广对象，建立了其到某类量子环面的代数同态，该同态是量子群上Feigin映射的$\boldsymbol{\tau}$类似物。
Affine $\imath$quantum groups and Steinberg varieties of type C
利用C型非连通Steinberg簇的等变K群，几何实现拟分裂仿射AIII$_{2n-1}^{(τ)}$型$\imath$量子群，并构造标准模与不可约模，给出标准模的合成重数公式。
Minimalistic Presentation and Coideal Structure of Twisted Yangians
本文通过Drinfeld型表示法引入扭曲杨代数的极简表示，并证明其作为杨代数的右余子代数结构，同时描述其在J表示下的同构性。
Shifted twisted Yangians of quasi-split ADE types
本文引入了偏移i阳离子代数，并展示了其与有限W代数的同构关系，探讨了其在仿射格拉斯曼切片固定点区域中的应用。
Braid group action and quasi-split affine $\imath$quantum groups II: higher rank
本文针对拟分裂仿射Satake图对应的量子对称对，利用相对辫子群作用构造泛$\imath$量子群的实、虚根向量及仿射秩一子代数，并据此建立Drinfeld型呈现。
$\imath$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 initiate a general approach to the relative braid group symmetries on (universal) $\imath$quantum groups, arising from quantum symmetric pairs of arbitrary finite types, and their modules. Our approach is built on new intertwining properties of quasi $K$-matrices which we develop and braid group symmetries on (Drinfeld double) quantum groups. Explicit formulas for these new symmetries on $\imath$quantum groups are obtained. We establish a number of fundamental properties for these symmetries on $\imath$quantum groups, strikingly parallel to their well-known quantum group counterparts. We ap…
$\imath$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。
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…
$\imath$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。
Quiver varieties and dual canonical bases
本文综述了量子群和$\imath$量子群对偶规范基理论的最新进展，介绍了$\imath$有向图代数，并展示了其在ADE类型准分裂$\imath$量子群中的两种实现方式，同时解决了若干猜想。
Dual canonical bases of quantum groups and $\imath$quantum groups I: Hall algebras
本文提出在$\imath$Hall 代数中构造$\imath$量子群的双重 canonical 基，并证明其在辫群作用和傅里叶变换下的不变性。
Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry
本文通过两种不同方法证明了$\imath$量子群的双canonical基的等价性，并展示了其结构常数的正性和Braiding群作用的不变性。
Shifted twisted Yangians and affine Grassmannian islices
本文研究了移位扭曲阳离子与仿射格拉斯曼流片的量子化关系，并提出框架用于生成库隆分支。
Twisted super Yangians of quasi-split type A
本文研究了准分裂型A的扭曲超Yangian代数，通过Gauss分解证明其与特殊扭曲超Yangian同构，并推广了非超情况下的已知结果。
Recently the authors initiated an $\imath$Hall algebra approach to (universal) $\imath$quantum groups arising from quantum symmetric pairs. In this paper we construct and study BGP type reflection functors which lead to isomorphisms of the $\imath$Hall algebras associated to acyclic $\imath$quivers. For Dynkin quivers, these symmetries on $\imath$Hall algebras induce automorphisms of universal $\imath$quantum groups, which are shown to satisfy the braid group relations associated to the restricted Weyl group of a symmetric pair; conjecturally these continue to hold for acyclic quivers/Kac-Mood…
A Drinfeld type presentation of twisted Yangians of quasi-split type
本文提出了一类扭曲杨代数的Drinfeld型呈现，并展示了其与反射代数的同构关系。
Compatibility of Drinfeld presentations and $q$-characters for affine Kac-Moody quantum symmetric pairs: quasi-split case
本文研究了准分裂仿形Kac-Moody量子对称对中Drinfeld级数的因子分解和余乘公式，并构造了与Frenkel和Reshetikhin的$ q $-字符同态相容的边界类比。
Compatibility of Drinfeld presentations for split affine Kac-Moody quantum symmetric pairs
本文证明分裂仿射Kac-Moody量子对称对中Drinfeld-Cartan算子的分解与余积公式，并证明边界$q$-特征映射与通常$q$-特征映射相容，同时给出扩展仿射Weyl群基本权的显式约化表达式。
Quasi-split iYangians: minimalistic presentations and coideal structures
本文研究拟分裂对称对关联的 iYangian，给出极简表示，构造到 Yangian 的嵌入并识别余理想，证明表示同构，并推导电流三角估计以分析限制权。
Shifted affine iquantum groups of quasi-split ADE types
本文通过Drinfeld呈现构建了任意准分裂ADE类型的移位仿射i量子群，并利用差分算子代数构造了GKLO型表示，从而构建了截断移位仿射i量子群，提供了截断移位iYangians的q变形。
Dual and double canonical bases of quantum groups
本文通过几何方法证明量子群的对偶典范基与 Berenstein--Greenstein 的双典范基一致，解决了若干关于正性和辫群作用不变性的猜想。
iQuantum groups and iHopf algebras II: dual canonical bases
本文基于i霍普夫代数构建了任意有限类型i量子群的对偶canonical基，并解决了Berenstein-Greenstein的若干猜想。
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