Andrei Neguţ
We consider the R-matrix of the quantum toroidal algebra of type gl_1, both abstractly and in Fock space representations. We provide a survey of a certain point of view on this object which involves the elliptic Hall and shuffle algebras, and show how to obtain certain explicit formulas.
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
Reduced quiver quantum toroidal algebras
本文给出了作用于非紧 toric Calabi-Yau 三维空间 BPS 态空间的约化箭图量子环代数的生成元-关系描述，并将其应用于得到对应带势箭图的 K-理论 Hall 代数模挠的描述。
Quantum toroidal algebras and solvable structures in gauge/string theory
本文综述了量子环面代数在二维共形场论、超对称规范理论与弦理论可解结构中的作用，详细探讨$\mathfrak{gl}_1$量子环面代数表示论及其在拓扑顶点与$qq$-字符等代数表述中的应用。
We introduce an algebra $\mathcal{K}_n$ which has a structure of a left comodule over the quantum toroidal algebra of type $A_{n-1}$. Algebra $\mathcal{K}_n$ is a higher rank generalization of $\mathcal{K}_1$, which provides a uniform description of deformed $W$ algebras associated with Lie (super)algebras of types BCD. We show that $\mathcal{K}_n$ possesses a family of commutative subalgebras.
Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian $\mathfrak{gl}_{1}$. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations. In this paper, we propose a $q$-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consist…
New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds
量子环面代数是由量子仿射代数进一步仿射化而得到的，并且和后者一样，可以用来构造可积系统。这些代数也描述了5D $\mathcal{N}=1$超对称规范理论的瞬子配分函数的对称性。本文考虑定义在轨道$S^1\times\mathbb{C}^2/\mathbb{Z}_p$上的规范理论，其中$\mathbb{Z}_p$的作用量由两个整数参数$(ν_1,ν_2)$决定.相应的量子环面代数被引入作为$\mathfrak{gl}(p)$的量子环面代数的变形。我们证明了它具有一个Hopf代数的结构，并提出了两个表示，称为垂直和水平，分别通过变形的Fock表示和斋藤的顶点表示的量子环形代数$\mathfrak{gl}(p)$。我们构建了这两种类型的表示之间交织的顶点运算符。这个对象被确定为一个$(ν_1,ν_2)$ -变形的细化拓扑顶点，使我们能够重建Nekrasov配分函数和$qq$ -字符的规范理论。
Quantum toroidal $\mathfrak{gl}_1$ algebra : plane partitions
在系列的第三篇论文中，我们构建了一个大家庭的表示的量子环形$\gl_1$代数，其基础是参数化的平面分区与各种边界条件和限制。我们研究了相应的形式特征。作为应用，我们得到了一类不可约最低权$\gl_\infty$ -模的Gelfand-Zetlin型基.
Twisted modules for Toroidal vertex algebras
这是系统研究环形顶点代数的系列论文中的一篇。在此之前，人们发展了一个关于环面顶点代数和模的理论，并将环面顶点代数明确地与环面李代数联系起来。本文研究了环面顶点代数的扭模。更具体地说，我们对具有有穷级自同构的一般环面顶点代数引入了扭模的概念，并给出了环面顶点代数和扭模的一般构造。然后，我们使用这种结构，建立一个自然的协会的toroidal顶点代数和扭曲的模扭曲toroidal李代数。这一结果与其他一些已知的结果一起意味着几乎所有的扩张仿射李代数都可以与环面顶点代数相关联。
Differential-difference system related to toroidal Lie algebra
我们提出了一个新的微分差分系统在（2+1）维时空（一个离散，两个连续），出现在Bogoyavlensky的（2+1）维KdV的层次。我们的方法是基于双线性恒等式的层次，这是有关的顶点算子表示的环形李代数$\sl_2^{tor}$。
Toroidal vertex algebras and their modules
本文发展了环面顶点代数及其模的理论，给出了环面顶点代数及其模的概念结构。作为应用，我们将环面顶点代数及其模与环面李代数联系起来。
Integrals of motion from quantum toroidal algebras
我们确定的转移矩阵的泰勒系数对应于量子环面代数的椭圆局部和非局部运动积分介绍的小岛，白石，渡边，和作者之一。这使我们能够在中长波模型上证明利特维诺夫定理。我们还讨论了XXZ模型的（gl（m），gl（n））对偶性及其对量子KdV模型的意义.特别是，我们猜想，谱的非局部积分的运动的Bazhanov，Lukyanov，Zamolodchikov的描述由Gaudin Bethe anomaly方程与仿射sl（2）。
Shuffle algebras, lattice paths and quantum toroidal $\mathfrak{gl}_{n|m}$
本文研究了$\mathfrak{gl}_{n|m}$矩阵洗牌代数中的通交元素，通过量子 affine 代数$U_t(\dot{\mathfrak{gl}}_{n|m})$的$R$-矩阵部分迹给出公式，并赋予格路解释。
Remarks on $q$-difference opers arising from quantum toroidal algebras
本文提出量子环面 $\mathfrak{gl}_2$ 代数 Bethe 谱与带表观奇点的 $q$-超几何 oper 之间的猜想对应，并引入、讨论 $q$-差分算子的表观奇点。
The deformed $\mathcal W$ algebras of type $\textsf{A}$ have a uniform description in terms of the quantum toroidal $\mathfrak{gl}_1$ algebra $\mathcal E$. We introduce a comodule algebra $\mathcal K$ over $\mathcal E$ which gives a uniform construction of basic deformed $\mathcal W$ currents and screening operators in types $\textsf{B},\textsf{C},\textsf{D}$ including twisted and supersymmetric cases. We show that a completion of algebra $\mathcal K$ contains three commutative subalgebras. In particular, it allows us to obtain a commutative family of integrals of motion associated with affine…
Vertex operator algebras and the representation theory of toroidal algebras
给出了环面李代数的一类不可约模的显式顶点算子代数构造。
Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials
本文研究了类型$\mathfrak{gl}_{1|1}$的位移量子扭代数，揭示了超Macdonald多项式与量子扭代数超荷作用之间的关系，并通过微分算子推导出Pieri规则，进一步推导出超对称哈密顿量，恢复了文献中的已有结果。
Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $ϕ$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $ϕ$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
In this paper, for every $ε\in \mathbb{Z}$, we introduce an extension of the 2-toroidal Lie algebra by certain derivations. Based on the $ϕ_ε$-coordinated modules theory for vertex algebras, we give an explicit realization of a class of irreducible highest weight modules for this extended toroidal Lie algebra. When $ε=1$, this affords a realization of certain irreducible modules for the toroidal extended affine Lie algebras first constructed by Billig.
Quantum N-toroidal algebras are generalizations of quantum affine algebras and quantum toroidal algebras. In this paper we construct a level-one vertex representation of the quantum N-toroidal algebra for type C. In particular, we also obtain a level-one module of the quantum toroidal algebra for type C as a special case.
Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization
本文引入量子N-环面代数作为量子环面代数的推广，并证明其为N重仿射化扩展量子GIM代数的商，推广了Berman-Moody定理。
In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.
We construct a level $-\frac{1}{2}$ vertex representation of the quantum N-toroidal algebra for type $C_n$, which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type $C_n$ as a by-product.
We introduce and study the quantum toroidal algebra $\mathcal{E}_{m|n}(q_1,q_2,q_3)$ associated with the superalgebra $\mathfrak{gl}_{m|n}$ with $m\neq n$, where the parameters satisfy $q_1q_2q_3=1$. We give an evaluation map. The evaluation map is a surjective homomorphism of algebras $\mathcal{E}_{m|n}(q_1,q_2,q_3) \to \widetilde{U}_q\,\widehat{\mathfrak{gl}}_{m|n}$ to the quantum affine algebra associated with the superalgebra $\mathfrak{gl}_{m|n}$ at level $c$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^{m-n}=c^2$. We also give a bosonic realization of level…
Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations. In this paper, we define the shifted QQTA and study a class of their representations. We define 1d a…
Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras
作为KP族理论的推广，讨论了（2+1）维非线性Schroedinger方程的族结构.给出了几种构造特解的方法。利用自由费米子的语言建立了环面李代数的表示与其族之间的关系。本文还讨论了与自对偶Yang-Mills方程的关系。
The $R$-matrix of the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ in the Fock module
提出了一种由Hopf代数中的余积关系计算Fock模张量积上的$R$ -矩阵$R$的方法.我们将这种方法应用到量子环形代数$U_{q,t}(\overset{..}{gl}_1)$，其中$R$目前还不明确。我们证明了$U_{q,t}(\overset{..}{gl}_1)$的余积关系简化为$R$的一个简单的方程。利用对称Macdonald多项式的理论，我们表明，这个方程提供了一个递归公式的矩阵元素的$R$。
We consider the problem of the R-matrix of the quantum toroidal algebra $U_{q,t}(\ddot{\mathfrak{gl}}_1)$ in the Fock representation. Using the connection between the R-matrix $R(u)$ ($u$ being the spectral parameter) and the theory of Macdonald operators we obtain explicit formulas for $R(u)$ in the operator and matrix forms. These formulas are expressed in terms of the eigenvalues of a certain Macdonald operator which completely describe the functional dependence of $R(u)$ on the spectral parameter $u$. We then consider the geometric R-matrix (obtained from the theory of K-theoretic stable b…
A Twisted Quantum Toroidal Algebra
作为量子TKK代数的一个类似，引入了A_1型扭曲量子环面代数。利用Fock空间上的扭曲量子顶点算子构造了新量子TKK代数的显式实现。
We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories $\mathscr{C}_1$ of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.
$U_q(\hat{\frak{gl}}_{{}_N})$ action on $\hat{\frak{gl}}_{{}_N}$ -modules and quantum toroidal algebras
通过量子广义线性代数构造了量子环面代数的顶点表示。利用量子广义线性代数的一种新实现，在仿射李代数$gl_n$的基本表示上构造了根向量的顶点算子，并证明了简单的生成元给出了双参数量子环面代数的一种实现.
Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras
设$X$是一个复流形，$π: E \rightarrow X$是一个具有纤维$F$的局部平凡全纯纤维化，而$\mathfrak{g}$是一个具有不变对称形式的李代数。我们关联到这个数据的一个全纯预分解代数$\mathcal{F}_{\mathfrak{g}, π}$上$X$在Costello-Gwilliam的形式主义。当$X=\mathbb{C}$，$\mathfrak{g}$是单的，$F$是光滑仿射簇时，我们从$\mathcal{F}_{\mathfrak{g}, π}$中抽取出一个顶点代数，它是李代数$\mathfrak{g} \otimes H^{0}(F, \mathcal{O})[z,z^{-1}]$的泛中心扩张的真空模.作为一种特殊情况，当$F$是一个代数环面$(\mathbb{C}^{*})^n$时，我们得到了一个自然地与一个$(n+1)$ --环面代数相联系的顶点代数，推广了仿射真空模.
Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions
本文利用量子环面 $\mathfrak{gl}_1$ 代数关系研究 $q,t$-矩阵模型的递归，统一证明并推广 Gaussian、Laguerre、Jacobi 等模型的 Macdonald 多项式超可积性公式。
Twisted quantum toroidal algebras $T_q^-(\mathfrak g)$
我们构造了Kac-Moody代数的一个主分次量子圈代数。作为一种特殊情况，与量子Serre关系一起得到了量子环形代数的扭曲模拟。
Finite dimensional modules over quantum toroidal algebras
量子环面代数的表示已被许多作者广泛研究。然而，当$q$是泛类时，还没有人构造出它们的有限维模.对所有的$\mathfrak{g}$ -泛$q$，若$\mathfrak{g}$不是$A_1$型，证明了量子环面代数$U_q(\mathfrak{g}_{\rm tor})$不存在非平凡的有限维单模.
Affine Yangians as Limits of Quantum Toroidal Algebras
本文证明了量子扭代数与非扭曲仿射杨代数之间存在退化同构，建立了仿射杨代数的PBW基和经典极限为多项式电流李代数的结构特性。
$(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
我们描述了提升Wess-Zumino-Witten模型从单圈Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$到一般量子环面代数的一般策略。一个几乎详尽的介绍给出了两个系列$U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$和$U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$，当筛选不存在，因此所有的分解是纯代数，即不包括额外的超几何型积分/求和。推广了Ding-Iohara-Miki（DIM）代数的交织算子（精化拓扑顶点）的构造，得到了$A_n$型量子环面代数的Fock表示的交织算子.这些算子的相关函数满足$(q,t)$ -Zamolodchikov（KZ）方程，该方程以${\cal R}$ -矩阵为特征.显式地计算了ALE空间上瞬子计数与Nekrasov函数的匹配。我们还提出了一个重要的应用DIM形式主义的研究$6d$规范理论描述的双椭圆可积系统。我们表明，规范理论的模块化和周期性的属性整齐地解释了网络矩阵模型提供解决方案的椭圆$(q,t)$ -KZ方程。
Fermionic realization of toroidal Lie algebras of classical types
我们使用费米子算子来构造经典类型的环面李代数，特别是辛仿射代数，这是第一次实现的费米子。
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…
Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal $\mathfrak{gl}_2$
本文构造 $\mathfrak{gl}_2$ 型量子环面代数的表示，得到生成元 $W_i(z)$，其代数结构为自由费米子代数与 $N=1$ 超 Virasoro 代数的直和的 $q$-形变，并证明其 screening 流与退化极限及二次关系。
We identify level one global Weyl modules for toroidal Lie algebras with certain twists of modules constructed by Moody-Eswara Rao-Yokonuma via vertex operators for type ADE and by Iohara-Saito-Wakimoto and Eswara Rao for general type. The twist is given by an action of $\mathrm{SL}_{2}(\mathbb{Z})$ on the toroidal Lie algebra. As a byproduct, we obtain a formula for the character of the level one local Weyl module over the toroidal Lie algebra and that for the graded character of the level one graded local Weyl module over an affine analog of the current Lie algebra.
Quantum toroidal algebras and their vertex representations
我们改变顶点表示的定义。因此，顶点表示只有一个参数。
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${\mathfrak {gl}}_n$ algebra ${\mathcal E}'_n(q_1,q_2,q_3)$ to the quantum affine algebra $U'_q\widehat{\mathfrak {gl}}_n$ at level $κ$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^n=κ^2$. We discuss ${\mathcal E}'_n(q_1,q_2,q_3)$ evaluation modules. We give highest weights of evaluation highest weight modules. We also obtain the decomposition of the evaluation Wakimoto module with respect to a Gelfand-Zeitlin type subalgebra of a completion of ${\mathcal E}'_n(q_1,q_2,q_3)$, which d…
In this paper, we prove that the quantum toroidal algebra of gl_n is isomorphic to the double shuffle algebra of Feigin and Odesskii for the cyclic quiver. The shuffle algebra viewpoint will allow us to prove a factorization formula for the universal R-matrix of the quantum toroidal algebra.
We introduce a new elliptic quantum toroidal algebra $U_{q,t,p}(gl_{1,tor})$. Various representations in the quantum toroidal algebra $U_{q,t}(gl_{1,tor})$ are extended to the elliptic case including the level (0,0) representation realized by using the elliptic Ruijsenaars difference operator. Intertwining operators of $U_{q,t,p}(gl_{1,tor})$-modules w.r.t. the Drinfeld comultiplication are also constructed. We show that $U_{q,t,p}(gl_{1,tor})$ gives a realization of the affine quiver $W$-algebra $W_{q,t}(Γ(\widehat{A}_0))$ proposed by Kimura-Pestun. This realization turns out to be useful to …
Bosonic realization of toroidal Lie algebras of classical types
推广Feingold-Frenkel的构造，我们利用Weyl玻色子场构造了分别为$-1, -2, -1/2$和-2水平的$A_n, B_n$、$C_n$和$D_n$型的环面李代数.特别地，在仿射李代数的情形下，我们的构造也给出了正交李代数的新的玻色构造。
Representations of the Quantum Toroidal Algebra on highest weight modules of the Quantum Affine Algebra of type gl(N)
在gl（N）型量子仿射代数的每个不可约可积最高权模上构造了sl（N）型量子Toroidal代数的表示.作为构造的中间步骤，我们得到了经典的能级-秩对偶的量子模拟，描述了q-Fock空间关于L级的U_q（gl（N）^）和N级的U_q（sl（L）^）的相互交换作用的互逆分解.
Energy-momentum tensor for the toroidal Lie algebras
我们构造了全（N+1）-环面李代数g的顶点算子表示.我们将g与一个环面顶点算子代数相关联，它是一个仿射VOA、一个双曲格VOA的子VOA、仿射sl（N）VOA和一个扭曲的Heisenberg-Virasoro VOA的张量积。环形VOA的模也是环形李代数g的模。我们还构造了对应于无发散向量场的环面李代数的一个重要子代数gdiv的不可约模。这个子代数带有一个非退化不变的双线性型。控制gdiv的表示论的VOA是在水平c处的仿射VOA Vaff（c）、双曲晶格VOA的子VOA、仿射sl（N）VOA和在水平cvir处的Virasoro VOA的张量积，其中关于中心电荷具有以下条件：2（N+1）+秩Vaff（c）+ cvir = 26。
Extremal loop weight modules and tensor products for quantum toroidal algebras
本文利用Drinfeld余积，用张量积定义了A型量子环面代数的可积表示.这使我们能够恢复最近由Feigin-Jimbo-Miwa-Mukhin [6]引入的向量表示，并由作者[21]构造为极值循环权重模块的子族。此外，我们得到了新的极值循环权模作为向量表示的张量幂幂的子项。作为一个应用，我们通过在单位根处特殊化量子参数，得到了量子环面代数的有限维表示。
Commutative subalgebra of a shuffle algebra associated with quantum toroidal $\mathfrak{gl}_{m|n}$
本文定义量子环面代数 $\mathcal E_{m|n}$ 的 shuffle 代数 $Sh_{m|n}$，构造并刻画一族参数化交换子代数，证明其为自由多项式代数并给出显式生成元。
Branching rules for quantum toroidal gl(n)
在量子环形代数的背景下构造了$Ugl(m+n)$的子代数$Ugl(n)\otimes Ugl(m)$的一个模拟，并研究了各种表示对这个子代数的限制.
Braid actions on quantum toroidal superalgebras
本研究证明了与gl_{m|n}型不同根系相关的量子环面超代数同构，且存在交换竖直与水平子代数的Miki自同构，核心方法是建立环面辫子群在这类代数直和上的作用。
Quantum toroidal algebras and motivic Hall algebras I. Hall algebras for singular elliptic curves
考虑算术亏格为$1$的不可约射影曲线上凝聚层的动力Hall代数.在奇异曲线情形下引入了复合子代数，并证明了它与光滑椭圆曲线的复合子代数同构。与Brurban和Schiffmann研究的光滑椭圆情形一样，复合子代数的约化Drinfeld二重代数同构于$\mathfrak{gl}_1$的量子环面代数（也称为Ding-Iohara-Miki代数），并且它继承了由相关导出范畴的等价导出的自同构。我们证明了一个非平凡的自同构与一个由三木在一个纯粹的代数方式。
Simple Toroidal Vertex Algebras and Their Irreducible Modules
本文继续了\cite{LTW}关于环面顶点代数的研究，研究了与环面李代数$L_{r}(\hat{\frak{g}})=\hat{\frak{g}}\otimes L_r$相联系的具体环面顶点代数，其中$\hat{\frak{g}}$是无扭仿射李代数，$L_r=$ \mathbb{C} [ t_{1} ^{ \pm 1}，\ldots，t_{r} ^{ \pm 1}] $. We first construct an $（r+1）$-toroidal vertex algebra $ V（T，0）$ and show that the category of restricted $ PDMATHAMY（\hat{\frak{g}}）$-modules is canonically isomorphic to that of $ V（T，0）$-modules.Let $ c $ denote the standard central element of $ \hat{\frak{g}} $ and set $ S_c=U（L_r（\mathbb{C} c））$. We furthermore study a distinguished subalgebra of $ V（T，0）$, denoted by $ V（S_c，0）$. We show that (graded) si…
We define an algebra $\mathcal{U}_0$ using a simplified set of generators for the quantum toroidal algebra $U_q(sl_{n+1}, tor)$ and show that there exists an epimorphism from $\mathcal{U}_0$ to $U_q(sl_{n+1}, tor)$. We derive a closed formula of the comultiplication on the generators of $\mathcal{U}_0$ that extends that of the quantum affine algebra $U_q(\hat{sl}_{n+1})$. As a consequence, we show that $\mathcal{U}_0$ is a Hopf algebra for $n=1, 2$ and give conjectural formulas in the general case. We further show that $\mathcal{U}_0$ is isomorphic to a double algebra.
Quantum toroidal algebras and their representations
利用量子仿射Kac-Moody代数的Drinfeld量子仿射化过程，定义了量子环面代数（或称双仿射量子代数）.它们是椭圆Cherednik代数（椭圆双仿射Hecke代数）的量子群类似物，它们通过Schur-Weyl对偶与之相关。在这篇评论文章中，我们给一个一瞥他们的非常丰富的表示论的背景下，一般量子仿射的某些方面。我们用几个例子来说明。我们还宣布新的结果，并解释可能的进一步发展，特别是在有限维表示根的统一。
Schur duality in the toroidal setting
本文的目的是用简单的代数方法证明GL（1）型Cherednik双仿射Hecke代数与V.Ginzburg，M. Kapranov和E. Vasserot在他们的“代数曲面上的朗兰兹对偶”。
We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$ algebra with a slope $n'/n$ using vertex operators of quantum affine $\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of an integrable level-one $U_q(\widehat{\mathfrak{gl}}_n)$-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Neguţ conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of $(\mathfrak{gl}_1,\mathfrak{gl}_n)$-duality.
Modules of the toroidal Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$
研究了双仿射李代数$\widehat{\widehat{\mathfrak{sl}}}_{2}$在一种新的三角分解下的最高权模.利用与水平仿射李子代数相似的条件确定了韦尔马模的奇异向量，并在$c_1>0$和$c_2=0$的条件下刻画了最高权模.
Little Strings, Quasi-topological Sigma Model on Loop Group, and Toroidal Lie Algebras
研究了A_{k-1} N=（2，0）小弦论的基态和左激发态.通过Atiyah [1]的一个定理，这些扇区可以被CP ^1上的超对称非线性西格玛模型捕获，目标空间是SU（k）的基环群。由L^2-上同调类描述的基态形成仿射李代数上的模，而由手征微分算子描述的左激发态形成环面李代数上的模。我们也将我们的结果应用于分析M5膜世界体积理论的1/2和1/4 BPS扇区。
Recently, Gaiotto and Rapcak proposed a generalization of $W_N$ algebra by considering the symmetry at the corner of the brane intersection (corner vertex operator algebra). The algebra, denoted as $Y_{L,M,N}$, is characterized by three non-negative integers $L, M, N$. It has a manifest triality automorphism which interchanges $L, M, N$, and can be obtained as a reduction of $W_{1+\infty}$ through a "pit" in the plane partition representation. Later, Prochazka and Rapcak proposed a representation of $Y_{L,M,N}$ in terms of $L+M+N$ free bosons through a generalization of Miura transformation, w…
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。