Samuel Buelk, Ben L. Cox, Elizabeth Jurisich
The authors construct a Wakimoto type realization of toroidal $\mathfrak{sl}_{n+1}$ The representation constructed in this paper utilizes non-commuting differential operators acting on the tensor product of two polynomial rings in many commuting variables.
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We prove the Kac-Wakimoto conjecture for the periplectic Lie superalgebra $\mathfrak{p}(n)$, stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.
Irreducibility of Certain $\widehat{\mathfrak{sl}}_2$-Modules of Wakimoto Type
本文研究最近构造的不可约光滑$\widehat{\mathfrak{sl}}_{2}$模，证明它们在临界和非临界水平下具有Wakimoto型实现，并推广了某些Wakimoto模，将其识别为广义Whittaker模。
We show that our construction of boson type realizations of affine $\mathfrak{sl}(n+1)$ in terms of intermediate Wakimoto modules gives representations that are generically isomorphic to certain Verma type modules. We then use this identification to obtain information about the submodule structure of intermediate Wakimoto modules
In this paper, we classify the irreducible integrable modules with finite dimensional weight spaces and non-trivial $\widetilde{\mathfrak g}_c$-action for the nullity $2$ toroidal extended affine Lie algebra $\widetilde{\mathfrak g}$, where $\widetilde{\mathfrak g}_c$ is the core of $\widetilde{\mathfrak g}$.
Free Field Realizations of Superelliptic Affine Lie Algebras
本文基于超椭圆仿射李代数的Wakimoto型自由场构造，研究sl_2的显式算子及偶奇sector特性，揭示自由场实现失败的统一障碍定理。
Representations of Toroidal and Full toroidal Lie algebras over polynomial algebras
本文研究环面与全环面李代数在秩1自由 $U(\tilde{\mathfrak{h}})$-模上的结构分类，证明其仅存在于 $A_l$、$C_l$ 型，并给出不可约条件。
Let $\mathbb{C}_q$ be a non-commutative Laurent polynomial ring associated with a $(n+1)\times (n+1)$ rational quantum matrix $q$. Let $\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ be the universal central extension of Lie subalgebra $\mathfrak{sl}_d(\mathbb{C}_q)$ of $\mathfrak{gl}_d(\mathbb{C}_q)$. Now let us take the Lie algebra $τ=\mathfrak{gl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$. Let $Der(\mathbb{C}_q)$ be the Lie algebra of all derivations of $\mathbb{C}_q$. Now we consider the Lie algebra $\tildeτ=τ\rtimes Der(\mathbb{C}_q)$, called as full toroidal Lie algebra co-ordin…
We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.
The fractional level models are (logarithmic) conformal field theories associated with affine Kac-Moody (super)algebras at certain levels $k \in \mathbb{Q}$. They are particularly noteworthy because of several longstanding difficulties that have only recently been resolved. Here, Wakimoto's free field realisation is combined with the theory of Jack symmetric functions to analyse the fractional level $\widehat{\mathfrak{sl}}(2)$ models. The first main results are explicit formulae for the singular vectors of minimal grade in relaxed Wakimoto modules. These are closely related to the minimal gra…
Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.
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.
We consider the (direct sum over all $n$ of the) $K$-theory of the semi-nilpotent commuting variety of $\mathfrak{gl}_n$, and describe its convolution algebra structure in two ways: the first as an explicit shuffle algebra (i.e. a particular $\mathbb{Z}[q_1^{\pm 1}, q_2^{\pm 1}]$-submodule of the equivariant $K$-theory of a point) and the second as the $\mathbb{Z}[q_1^{\pm 1}, q_2^{\pm 1}]$-algebra generated by certain elements $\{\bar{H}_{n,d}\}_{(n,d) \in \mathbb{N} \times \mathbb{Z}}$.
(Affine) $\mathcal{W}$-algebras are a family of vertex algebras defined by the generalized Drinfeld-Sokolov reductions associated with a finite-dimensional reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, a nilpotent element $f$ in $[\mathfrak{g},\mathfrak{g}]$, a good grading $Γ$ and a symmetric invariant bilinear form $κ$ on $\mathfrak{g}$. We introduce free field realizations of $\mathcal{W}$-algebras by using Wakimoto representations of affine Lie algebras, where $\mathcal{W}$-algebras are described as the intersections of kernels of screening operators. We call these Wakimoto free …
This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of $\mathfrak{sl}_n$, denoted by $\mathcal{U}^{(n)}_{q_1,q_2,q_3}$ and $\mathcal{Y}^{(n)}_{h_1,h_2,h_3}$, respectively. Our motivation arises from the milestone work of Gautam and Toledano Laredo, where a similar relation between the quantum loop algebra $U_q(L \mathfrak{g})$ and the Yangian $Y_h(\mathfrak{g})$ has been established by constructing an isomorphism of $\mathbb{C}[[\hbar]]$-algebras $Φ:\widehat{U}_{\exp(\hbar)}(L\mathfrak{g})\to \widehat{Y}_\hbar(\mathfrak{g})$ (with $\ \widehat{}\ $ sta…
Shuffle approach towards quantum affine and toroidal algebras
本文系统讲解 shuffle 代数方法在量子环路代数与量子环面代数中的应用，通过特殊化映射构造新 PBWD 基、Fock 模的几何解释、Heisenberg 代数作用及 Bethe 交换子代数的显式计算。
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…
Combinatorial bases in quantum toroidal $\mathfrak{gl}_2$ modules
本文通过 $q$-特征理论，以组合方式构造量子扭棱镜 $\mathfrak{gl}_2$ 模中的基，并展示了评价模块的构造方法及彩色平面分割的应用。
Let $X$ be a complex manifold, $π: E \rightarrow X$ a locally trivial holomorphic fibration with fiber $F$, and $\mathfrak{g}$ a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra $\mathcal{F}_{\mathfrak{g}, π}$ on $X$ in the formalism of Costello-Gwilliam. When $X=\mathbb{C}$, $\mathfrak{g}$ is simple, and $F$ is a smooth affine variety, we extract from $\mathcal{F}_{\mathfrak{g}, π}$ a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra $\mathfrak{g} \otimes H^{0}(F, \mathcal{O})[z,z^{-1}]$…
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。
We define the double quantum affinization $\ddot{\mathrm{U}}_q(\mathfrak a_1)$ of type $\mathfrak{a}_1$ as a topological Hopf algebra. We prove that it admits a subalgebra $\ddot{\mathrm{U}}_q'(\mathfrak a_1)$ whose completion is (bicontinuously) isomorphic to the completion of the quantum toroidal algebra $\dot{\mathrm{U}}_q(\dot{\mathfrak a}_1)$, defined as the (simple) quantum affinization of the untwisted affine Kac-Moody Lie algebra $\dot{\mathfrak{sl}}_2$ of type $\dot{\mathfrak a}_1$, equipped with a certain topology inherited from its natural $\mathbb Z$-grading. The isomorphism is con…
Folding of cluster algebras and quantum toroidal algebras
本文通过簇理论解释量子扭代数中$q$-字符的折叠映射，并证明了相关猜想，同时探讨了特定模的$q$-字符性质。
Elliptic Quantum Toroidal Algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ and Elliptic Stable Envelopes for the $A^{(1)}_{N-1}$ Quiver Varieties
本文提出构造椭圆量子环面代数顶点算子的新方法，利用椭圆稳定包络公式，所得算子满足交换关系并构造L-算子，证明其为模的 intertwining 算子。
Let $L$ be the basic (level one vacuum) representation of the affine Kac-Moody Lie algebra $\hat{\mathfrak g}$. The $m$-th space $F_m$ of the PBW filtration on $L$ is a linear span of vectors of the form $x_1... x_lv_0$, where $l\le m$, $x_i\in \hat{\mathfrak g}$ and $v_0$ is a highest weight vector of $L$. In this paper we give two descriptions of the associated graded space $L^{\rm gr}$ with respect to the PBW filtration. The "top-down" description deals with a structure of $L^{\rm gr}$ as a representation of the abelianized algebra of generating operators. We prove that the ideal of relatio…
We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the $N$--fold fusion product of $V(λ)$. This result shows that the fusion product of these types is independent of the chosen parameters, proving a special case of a conjecture by Fei…
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。
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。
The PBW basis of $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$
本文研究 A 型量子环面代数 $U_{q,\bar{q}}(\ddot{\mathfrak{gl}}_n)$ 的 PBW 基，证明其生成元具有类似 n=1 情形的交换关系，并给出该代数的新表示。
We discuss the 5d AGT correspondence of supergroup gauge theories with A-type supergroups. We introduce two intertwiners called positive and negative intertwiners to compute the instanton partition function. The positive intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the negative intertwiner is an intertwiner obtained by using central charges with negative levels. We show that composition of them gives the basic Nekrasov factors appearing in supergroup partition functions. We explicitly derive the instanton partition functions of supergroup gauge theories with A and D-typ…
We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody $U_q(\widehat{\mathfrak{g}})_k$ to generic quantum toroidal algebras. A nearly exhaustive presentation is given for the two series $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ and $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_n)$, when screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations. Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (…
Generalized finite and affine $W$-algebras in type $A$
本文提出了一种基于BRST复形的构造方法，生成了一类新的广义有限和仿射W代数，并展示了其在顶点代数中的应用。
Automorphisms of quantum toroidal algebras from an action of the extended double affine braid group
本文构造扩展双仿射辫群对量子环面代数的作用，并利用其产生交换水平和垂直子代数的自同构，推广了 Miki 在 A 型中的结果。
Representations of quantum toroidal superalgebras and plane $\mathbf{s}$-partitions
本文构造了关联$\frak{gl}_{m|n}$李超代数与宇称$\text{s}$的量子环面超代数$\frak{E}_\text{s}$的Fock模与MacMahon模，明确其基的标记方式与生成元作用的Pieri型公式，并研究了对应特征标。
Orthosymplectic Feigin-Semikhatov duality
本文研究了正交辛类型的Feigin-Semikhatov对称性，通过相对半无限上同调和能级流动扭，建立了W超代数权重模的块等价性，并推导了特征公式。
Rational, trigonometric, elliptic algebras and moduli spaces of sheaves on surfaces
本文综述了Yangian/量子环面代数在曲面稳定层模空间上的作用，并推广至椭圆上同调。
Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras
针对所有非扭型量子环面代数，引入基于反对合扭曲Drinfeld余积的新拓扑余积，定义可积表示范畴的张量积，证明其与Drinfeld多项式兼容等仿射层结果的量子环面类似物，还得到解Yang-Baxter方程的$R$-矩阵。
Athinization of irreducible $\widehat{\mathfrak{gl}}_n$-modules with dominant highest weights
研究通过薄化方法将不可约 $\widehat{\mathfrak{gl}}_n$-模块转化为 affine Yangian 上的薄模块，提供新的组合实现。
We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level 1 representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.
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