Xin Fang
We give a unified construction of quantum groups, q-Boson algebras and quantized Weyl algebras and an action of quantum groups on quantized Weyl algebras. This enables us to give a conceptual proof of the semi-simplicity of the category $\mathcal{O}(B_q)$ introduced by T.Nakashima and the classification of all simple objects in it.
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Belov-Kanel and Kontsevich conjectured that the group of automorphisms of the n'th Weyl algebra and the group of polynomial symplectomorphisms of C^2 are canonically isomorphic. We discuss how this conjecture can be approached by means of (second) quantized Weyl algebras at roots of unity.
Weyl group extension of quantized current algebras
在本文中，我们在几个方向上扩展量子仿射代数$U_q(\hat {\gg})$和Yangians的Drinfeld电流实现：我们构造了${\gg}$的非单根的电流算子，根据当前算子定义一个新的辫群作用，并描述相应“”的泛R-矩阵。关于流算子的无穷乘积形式和某些积分形式的Drinfeld "余乘。
Simple Modules Over Quantized Weyl Algebras At Roots Of Unity
该研究完成了代数闭域上秩二单位根量子化Weyl代数的单模分类工作。
In this note we solve the isomorphism problem for the multiparameter quantized Weyl algebras, in the case when none of the deformation parameters q_i is a root of unity, over an arbitrary field.
We prove an analogue of Weyl's law for quantized irreducible generalized flag manifolds. By this we mean defining a zeta function, similarly to the classical setting, and showing that it satisfies the following two properties: as a functional on the quantized algebra it is proportional to the Haar state; its first singularity coincides with the classical dimension. The relevant formulae are given for the more general case of compact quantum groups.
We prove an integral version of the Schur--Weyl duality between the specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006), 138--174] in the symplectic case. As a byproduct, we show that, as $Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is…
In this paper, we study the algebra automorphisms and isomorphisms for a family of "symmetric" multiparameter quantized Weyl algebras $\A$ and some related algebras in the generic case. First, we compute the Nakayama automorphism for $\A$ and give a necessary and sufficient condition for $\A$ to be Calabi-Yau. We also prove that $\A$ is cancellative. Then we determine the automorphism group for $\A$ and its polynomial extension $\E$. As an application, we solve the isomorphism problem for $\{\A\}$ and $\{\E\}$. Similar results will be established for the Maltisiniotis multiparameter quantized …
We classify the centers of the quantized Weyl algebras that are PI and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this …
For a quantized enveloping algebra of a complex semisimple Lie algebra with deformation parameter not a root of unity, we classify all homogeneous right coideal subalgebras. Any such right coideal subalgebra is determined uniquely by a triple consisting of two elements of the Weyl group and a subset of the set of simple roots satisfying some natural conditions. The essential ingredients of the proof are the Lusztig automorphisms and the classification of homogeneous right coideal subalgebras of the Borel Hopf subalgebras of quantized enveloping algebras obtained previously by H.-J. Schneider a…
Let R be a Noetherian domain and let (σ, δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F = t^2 - v is normal in the Ore extension R[t; σ, δ] where v ε R. We show F is prime in R[t; σ, δ] if and only if F is irreducible in Q[t; σ, δ] if and only if there does not exist w ε Q such that v = σ (w)w - δ (w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.
We shall construct the quantized q-analogues of the birational Weyl group actions arising from nilpotent Poisson algebras, which are conceptual generalizations, proposed by Noumi and Yamada, of the Bäcklund transformations for Painlevé equations. Consider a quotient Ore domain of the lower nilpotent part of a quantized universal enveloping algebra of arbitrary symmetrizable Kac-Moody type. Then non-integral powers of the image of the Chevalley generators generate the quantized q-analogue of the birational Weyl group action. Using the same method, we shall reconstruct the quantized Bäcklund tra…
Bell and Zhang have shown that if $A$ and $B$ are two connected graded algebras finitely generated in degree one that are isomorphic as ungraded algebras, then they are isomorphic as graded algebras. We exploit this result to solve the isomorphism problem in the cases of quantum affine spaces, quantum matrix algebras, and homogenized multiparameter quantized Weyl algebras. Our result involves determining the degree one normal elements, factoring out, and then repeating. This creates an iterative process that allows one to determine relationships between relative parameters.
Dynamical Weyl groups and applications
继前面的论文Tarasov和第二作者，我们定义和研究一个新的结构，这可能被视为动力学模拟的Weyl群的李代数和量子Weyl群的量子包络代数。我们给出了这种新结构的一些应用。
For an algebraically closed field $K$, we investigate a class of noncommutative $K$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots,x_n\}$ such that each pair satisfies a relation of the form $x_ix_j=q_{ij}x_jx_i+r_{ij}$, where $q_{ij}\in K^*$ and $r_{ij}\in K$, with, in some sense, sufficiently many pairs for which $r_{ij}\neq 0$. We classify connected quantized Weyl algebras, showing that there are two types, linear and cyclic, each depending on a single parameter $q$. When $q$ is not a root of unity we determine the pri…
We show that the Schur-Weyl type duality between $gl(1|1)$ and $GL_n$ gives a natural representation-theoretic setting for the relation between reduced and non-reduced Burau representations.
We compute the center and Azumaya locus in the simplest non-abelian examples of quantized multiplicative quiver varieties at a root of unity: quantum Weyl algebras of rank $N$, and quantum differential operators on the quantum group $\mathrm{GL}_2$. These examples illustrate in elementary terms much more general phenomena explored further in [Ganev-Jordan-Safronov 2019].
For the Borel part of a quantized enveloping algebra we classify all right coideal subalgebras for which the intersection with the coradical is a Hopf algebra. The result is expressed in terms of characters of the subalgebras $U^+[w]$ of the quantized enveloping algebra, introduced by de Concini, Kac, and Procesi for any Weyl group element $w$. We explicitly determine all characters of $U^+[w]$ building on recent work by Yakimov on prime ideals of $U^+[w]$ which are invariant under a torus action. Key words: Quantum groups, coideal subalgebras
Dual partially harmonic tensors and quantized Schur--Weyl duality
本文通过图示范畴和标准基，证明了辛代数与Birman-Murakami-Wenzl代数之间的量化Schur-Weyl对偶性。
Recently, Kuniba, Okado and Yamada proved that the transition matrix of PBW-type bases of the positive-half of a quantized universal enveloping algebra $U_q(\mathfrak{g})$ coincides with a matrix coefficients of the intertwiner between certain irreducible modules over the corresponding quantized coordinate ring $A_q(\mathfrak{g})$, introduced by Soibelman. In the present article, we give a new proof of their result, by using representation theory of the $q$-boson algebra, and the Drinfeld paring of $U_q(\mathfrak{g})$.
We derive a family of solutions to the tetrahedron equation using the RTT presentation of a two parametric quantized algebra of regular functions on an upper triangular subgroup of GL(n). The key ingredients of the construction are the longest element of the Weyl group, the quantum dilogarithm function, and central elements of the quantized division algebra of rational functions on the subgroup in question.
Quantum Heisenberg--Weyl Algebras
对Heisenberg-Weyl代数$[A_+,A_-]=M$上的所有李双代数结构进行了分类和量子化.这样得到的量子Heisenberg-Weyl代数的完整列表包括新的多参数变形，其中大多数是非共边界型的。
Let $R$ be a commutative ring with one and $q$ an invertible element of $R$. The (specialized) quantum group ${\mathbf U} = U_q(\mathfrak{gl}_n)$ over $R$ of the general linear group acts on mixed tensor space $V^{\otimes r}\otimes {V^*}^{\otimes s}$ where $V$ denotes the natural $\mathbf U$-module $R^n$, $r,s$ are nonnegative integers and $V^*$ is the dual $\mathbf U$-module to $V$. The image of $\mathbf U$ in $\mathrm{End}_R(V^{\otimes r}\otimes {V^*}^{\otimes s})$ is called the rational $q$-Schur algebra $S_{q}(n;r,s)$. We construct a bideterminant basis of $S_{q}(n;r,s)$. There is an actio…
Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan-Weyl realizations for quantum affine algebras
我们表明，一些因素的通用R-矩阵产生家庭的扭曲的标准的Hopf结构的任何量化contragradient李（超）代数的有限增长。作为应用，我们证明了任意两个具有不同Cartan矩阵的同构超代数具有同构的q-变形（作为结合超代数），并且它们的标准余乘是通过这种扭曲连接的。我们还提出了量子化仿射非扭曲卡茨-穆迪代数的第二Drinfeld实现的生成元与Cartan-Weyl生成元之间的显式关系。讨论了与这种同构密切相关的量子Cartan-Weyl基理论的进一步发展。证明了第二种实现的一个余乘的Drinfeld公式是标准的余乘被泛R-矩阵的因子扭曲的结果。最后，讨论了Drinfeld余乘的性质.
Dimensions of quantized tilting modules
设$U$是素数$p$在$p-$次单位根上具有整除幂的量子群.对于相应的仿射Weyl群中的任何双边胞腔$A$，人们将张量理想关联到$U$上的倾斜模范畴中。本文证明了对于任意胞元$A$，存在从相应的张量理想倾斜的模$T$，使得$p$整除$dim T$的最大幂是$p^{a(A)}$，其中$a(A)$是Lusztig的$a-$函数.在新的版本中，一些错别字被纠正，并根据裁判的建议改进了说明。
Regions of linearity, Lusztig cones and canonical basis elements for the quantized enveloping algebra of type A_4
设U_q是秩为n的李代数g的量子群. U的负部分U^-有一个正则基B，它具有良好的性质，由Kashiwara和Lusztig引入。Kashiwara和Lusztig的方法导致了一组典型基的替代参数化，一个用于g的外尔群中最长词的每个简化表达式。我们证明了当g为A_4型时，Lusztig锥、正则基元和由上述参数化所产生的重参数化函数的线性区域之间有密切的关系。图可以定义在关于邻接的线性单纯区域集合上，并且我们进一步证明了该图与具有由外尔群最长字模交换的约化表达式给出的顶点和由长辫子关系给出的边的图同构。关键词：量子群，李代数，正则基，紧单项式，外尔群，分段线性函数。
Twists of twisted generalized Weyl algebras
本文研究扭曲广义外尔代数的分次扭曲张量积与分次扭转，证明该类在温和假设下封闭，并将多参数量子化外尔代数的余循环等价结果推广到TGWAs，进而证明某些A₂型TGWAs是诺特的。
We prove that every endomorphism of a simple quantum generalized Weyl algebra $A$ over a commutative Laurent polynomial ring in one variable is an automorphism. This is achieved by obtaining an explicit classification of all endomorphisms of $A$. Our main result applies to minimal primitive factors of the quantized enveloping algebra of $U_q(\mathfrak{sl}_2)$ and certain minimal primitive quotients of the positive part of $U_q(\mathfrak{so}_5)$.
Lowering operators, orthogonal decomposition of tensor space, and quantized Schur--Weyl duality
本文通过递归构造某些线性组合Ψ_j，给出了张量空间V_q^⊗r的异型半单分解的组合实现，扩展了之前关于n=2的情况的研究。
Let $\mathfrak{g}$ be a hyperbolic Kac-Moody algebra of rank $2$. We give a polyhedral realization of the crystal basis for the extremal weight module of extremal weight $λ$, where $λ$ is an integral weight whose Weyl group orbit has neither a dominant integral weight nor an antidominant integral weight.
Locally finite simple weight modules over twisted generalized Weyl algebras
给出了描述扭曲广义Weyl代数上单权模的方法和显式公式。当某个交换子代数在代数闭域上生成时，我们得到了一类局部有限单权模的分类，它们是由同构于非交换环面的张量积的子代数上的单模导出的.作为一个应用，我们描述了量子化Weyl代数上的简单权模.
The Lusztig cones of a quantized enveloping algebra of type A
证明了对于A_n型外尔群中最长字的每一个约化表达式，Lusztig用紧单项式描述正则基时所对应的锥是单纯的，并构造了显式的生成向量.
In this paper, we establish explicit relationship between decomposition numbers of quantized walled Brauer algebras and those for either Hecke algebras associated to certain symmetric groups or (rational) $q$-Schur algebras over a field $κ$. This enables us to use Ariki's result \cite{Ar} and Varagnolo-Vasserot's result \cite{VV} to compute such decomposition numbers via inverse Kazhdan-Lusztig polynomials associated with affine Weyl groups of type $A$ if the ground field is $\mathbb C$.
We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter …
New solutions to the tetrahedron equation associated with quantized six-vertex models
本文构造了与量子化六顶点模型相关的一族四面体方程新解，分别基于q-振子代数和q-Weyl代数，后者给出可因式分解或表为截断q-超几何级数的新R矩阵。
We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremalweight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.
We prove that any twisted generalized Weyl algebra satisfying certain consistency conditions can be embedded into a crossed product. We also introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordan's simple localization of the multiparameter quantized Weyl algebra and Hayashi's $q$-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus,…
Quantized Weyl algebras, the double centralizer property, and a new First Fundamental Theorem for $U_q(\mathfrak{gl}_n)$
本文定义了矩阵空间量子化坐标环上的多项式系数微分算子代数的$q$-模拟，证明两个量子包络代数像与其交集互为互中心化子，从而建立了$U_q(\mathfrak{gl}_n)$不变理论的新第一基本定理，并给出Cartan子代数像交集的$q$-行列式生成元公式。
We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the a…
The prime and primitive spectra of multiparameter quantum symplectic and euclidean spaces
我们研究了一类代数，它提供了量子辛空间和量子欧氏$2n$ -空间的多参数版本。这些代数包括分次量子化Weyl代数，量子化海森堡空间，以及一类代数介绍了哦。我们描述了这些代数的素理想和本原理想的结构。其他结构结果包括正常分离和链状。
Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions
我们研究了流代数、圈代数及其量子化形式的有限维表示。对于{ \tt ADE}型单李代数的当前代数，我们证明了Kirillov-Reshetikhin模和Weyl模实际上都是Demazure模.因此，人们得到了一个初等证明的维数公式的Weyl模的电流和循环代数。此外，我们表明，晶体的外尔和Demazure模块是相同的一些额外的标签零箭头外尔模块。对于任意单李代数的当前代数$\Lgc$，同一层的Demazure模的融合积又是一个Demazure模.作为应用，我们构造了Kac-Moody代数$\Lhg$ -模的$\Lgc$ -模结构，即有限维$\Lgc$ -模的半无限融合积$V(\ell\Lam_0)$.
Drinfel'd doubles and Shapovalov determinants
计算了一类点霍普夫代数的Shapovalov行列式，包括量子化包络代数、Lusztig小量子群和量子化李超代数.我们的主要工具是根系，Weyl群胚，和Lusztig型同构。我们阐述了强大的新技术的代数在根的单位，并通过一般情况下使用的密度参数。关键词：霍普夫代数，尼科尔斯代数，量子群，表示
We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.
Motives of Brauer-Severi schemes of Cayley-smooth algebras associated to homogeneous superpotentials are used to compute inductively the motivic Donaldson-Thomas invariants of the corresponding Jacobian algebras. This approach can be used to test the conjectural exponential expressions for these invariants, proposed in arXiv:1510.08116. As an example we confirm the second term of the conjectured expression for the motivic series of the homogenised Weyl algebra.
We construct homomorphic images of $su(n,n)^{\mathbb C}$ in Weyl Algebras ${\mathcal H}_{2nr}$. More precisely, and using the Bernstein filtration of ${\mathcal H}_{2nr}$, $su(n,n)^{\mathbb C}$ is mapped into degree $2$ elements with the negative non-compact root spaces being mapped into second order creation operators. Using the Fock representation of ${\mathcal H}_{2nr}$, these homomorphisms give all unitary highest weight representations of $su(n,n)^{\mathbb C}$ thus reconstructing the Kashiwara--Vergne List for the Segal--Shale--Weil representation. Just as in the derivation of the their l…
We prove a result for the commutator of quantum root vectors corresponding to cominuscole parabolics. Specifically we show that, given two quantum root vectors, belonging respectively to the quantized nilradical and the quantized opposite nilradical, their commutator belongs to the quantized Levi factor. This generalizes the classical result for Lie algebras. Recall that the quantum root vectors depend on the reduced decomposition of the longest word of the Weyl group. We show that this result does not hold for all such choices. We conjecture that it holds when the reduced decomposition is app…
Quantized semisimple Lie groups
本文从酉表示理论角度介绍了半单李群的q-形变，通过聚焦$\mathrm{sl}(2,\mathbb{C})$及其相关李群，探讨了量子群、量子化代数及Plancherel公式量子类似物等核心主题。
On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts
研究$\mathbb{Z}$-分级顶点代数的模式转换代数及其在玻色鬼模块中的应用，证明其弱锁性质及构造更高层次朱代数。
Unitarizability of Harish-Chandra bimodules over generalized Weyl and $q$-Weyl algebras
本文对广义Weyl和$q$-Weyl代数上的Harish-Chandra双模，在一般量子化参数下分类了不变正定Hermite形式，从而解决其单位化性问题。
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new represent…
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