Vertex operator algebras and the representation theory of toroidal algebras

Bosonic vertex representations of the toroidal superalgebras in type D(m,n)

In this paper, vertex representations of the 2-toroidal Lie superalgebras of type $D(m, n)$ are constructed using both bosonic fields and vertex operators based on their loop algebraic presentation.

Toroidal extended affine Lie algebras and 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.

Representations of Twisted Toroidal Lie Algebras from Twisted Modules over Vertex Algebras

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.

Irreducible modules of toroidal Lie algebras arising from $ϕ_ε$-coordinated modules of vertex 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 toroidal algebras and their vertex representations

We change the definition of the vertex representations. As a result the vertex representations has one parameter.

Energy-momentum tensor for the toroidal Lie algebras

We construct vertex operator representations for the full (N+1)-toroidal Lie algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl(N) VOA and a twisted Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal Lie algebra g. We also construct irreducible modules for an important subalgebra gdiv of the toroidal Lie algebra that corresponds to the divergence free vector fields. This subalgebra carries a non-degenerate invariant bilinear form. The VOA that co…

Twisted quantum toroidal algebras $T_q^-(\mathfrak g)$

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.

Fermionic realization of toroidal Lie algebras of classical types

We use fermionic operators to construct toroidal Lie algebras of classical types, including in particular that of symplectic affine algebras, which is first realized by fermions.

Level one Weyl modules for toroidal Lie algebras

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.

Level $-1/2$ realization of quantum N-toroidal algebras in type $C_n$

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.

Bosonic realization of toroidal Lie algebras of classical types

Generalizing Feingold-Frenkel's construction we use Weyl bosonic fields to construct toroidal Lie algebras of types $A_n, B_n$, $C_n$ and $D_n$ of level $-1, -2, -1/2$ and -2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.

Extremal loop weight modules and tensor products for quantum toroidal algebras

We define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld "coproduct". This allow us to recover the vector representations recently introduced by Feigin-Jimbo-Miwa-Mukhin [6] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.

Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras

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 $\mathfrak{g} \otimes H^{0}(F, \mathcal{O})[z,z^{-1}]$…

On Hopf algebraic structures of quantum toroidal algebras

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

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich representation theory in the context of general quantum affinizations. We illustrate with several examples. We also announce new results and explain possible further developments, in particular on finite dime…

与量子环面$\boldsymbol{\frak{gl}}_N$相关的量子顶点代数

Quantum vertex algebra associated to quantum toroidal $\mathfrak{gl}_N$

本研究通过等变φ-坐标拟模，形变泛仿射顶点代数构造出ℏ-进量子Z-顶点代数，证明水平为ℓ的限制gl_N型量子环面代数模的范畴与该量子顶点代数的等变拟模范畴典范同构，建立了二者的关联。

$U_q(\hat{\frak{gl}}_{{}_N})$ action on $\hat{\frak{gl}}_{{}_N}$ -modules and quantum toroidal algebras

We construct a vertex representation for the quantum toroidal algebra through the quantum general linear algebra. Using a new realization of the quantum general linear algebra we construct vertex operators for root vectors on the basic representation of the affine Lie algebra $gl_n$ and show that the simple generators give rise a realization of the quantum toroidal algebra with two parameters.

Modules of the toroidal Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$

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 $c_2=0$.

Toroidal Grothendieck rings and cluster algebras

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.

椭圆曲面与环面超代数

Elliptic surfaces and toroidal superalgebras

本文利用环面扩展仿射李代数的顶点表示和 bc-ghost 系统的几何解释，证明了等平凡椭圆曲面上层模空间的等变上同调具有无穷维李（超）代数表示，并探讨了其在枚举几何中的应用。

Quantum toroidal and shuffle algebras

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.

量子N-环面代数与N重仿射化的扩展量子GIM代数

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。

On quantum toroidal algebra of type $A_1$

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.

椭圆量子环面代数 U_{t_1,t_2,p}(gl_{1,tor})、顶点算子与 L-算子

Elliptic Quantum Toroidal Algebra U_{t_1,t_2,p}(gl_{1,tor}), Vertex Operators and L-operators

本文提出椭圆量子环面代数的两类新顶点算子，其复合重现椭圆稳定包络的混洗积公式，并构造满足 RLL=LLR^* 关系的 L-算子，从而定义余乘法，证明顶点算子为交缠算子。

The algebra $U_q(\hat{sl}_\infty)$ and applications

In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras, more precisely predictions of character formulae for representations of quantum toroidal algebras. In this context, we state a general positivity conjecture for representations of $U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras, that we prove for Kirillov-Reshetikhin modules.

簇代数与量子扭代数的折叠

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。

A Twisted Quantum Toroidal Algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A_1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.

Finite dimensional modules over quantum toroidal algebras

The representations of the quantum toroidal algebras have been widely studied by many authors. However, no one has constructed some finite dimensional modules for them while $q$ is generic. In this paper, for all $\mathfrak{g}$-generic $q$, if $\mathfrak{g}$ is not of type $A_1$, we prove that the quantum toroidal algebra $U_q(\mathfrak{g}_{\rm tor})$ has no nontrivial finite dimensional simple module.

An integral form of quantum toroidal $\mathfrak{gl}_1$

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 $\{\bar{H}_{n,d}\}_{(n,d) \in \mathbb{N} \times \mathbb{Z}}$.

椭圆量子环面代数 $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ 与 $A^{(1)}_{N-1}$ 箭图簇的椭圆稳定包络

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。

Twisted vertex representations and spin characters

We establish a new group-theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras of ADE types in the same spirit of our new approach to the McKay correspondence. Our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group.

Representations of the Quantum Toroidal Algebra on highest weight modules of the Quantum Affine Algebra of type gl(N)

A representation of the Quantum Toroidal Algebra of type sl(N) is constructed on every irreducible integrable highest weight module of the Quantum Affine Algebra of type gl(N). As an intermediate step in the construction, we obtain a quantum analogue of the classical level-rank duality, describing the reciprocal decomposition of the q-Fock space with respect to mutually commutative actions of U_q(gl(N)^) of level L, and U_q(sl(L)^) of level N.

Toroidal and Elliptic Quiver BPS Algebras and Beyond

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…

量子环面代数与规范/弦理论中的可解结构

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。

Quantum vertex representations via finite groups and the McKay correspondence

We establish a $q$-analog of our recent work on vertex representations and the McKay correspondence. For each finite group $Γ$ we construct a Fock space and associated vertex operators in terms of wreath products of $Γ\times \mathbb C^{\times}$ and the symmetric groups. An important special case is obtained when $Γ$ is a finite subgroup of $SU_2$, where our construction yields a group theoretic realization of the representations of the quantum affine and quantum toroidal algebras of $ADE$ type.

K-theoretic Hall algebras, quantum groups and super quantum groups

We first prove that the K-theoretic Hall algebra of a preprojective algebra of affine type is isomorphic to the positive half of a quantum toroidal quantum group. An essential step consists to deform the K-theoretic Hall algebra so that the deformation is torsion free over some polynomial subalgebra. Next, we compare super toroidal quantum groups of type A with K-theoretic Hall algebras of quivers with potential, which are defined using the Grothendieck groups of categories of singularities of some Landau-Ginzburg models.

New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(ν_1,ν_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathf…

The $R$-matrix of the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ in the Fock module

We propose a method to compute the $R$-matrix $R$ on a tensor product of Fock modules from coproduct relations in a Hopf algebra. We apply this method to the quantum toroidal algebra $U_{q,t}(\overset{..}{gl}_1)$ for which $R$ is currently not explicitly known. We show that the coproduct relations of $U_{q,t}(\overset{..}{gl}_1)$ reduce to a single elegant equation for $R$. Using the theory of symmetric Macdonald polynomials we show that this equation provides a recursive formula for the matrix elements of $R$.

Two-parameter quantum vertex representations via finite groups and the McKay correspondence

We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of $SL_2(\mathbb C)$ via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of $ADE$ types.

The R-matrix of the quantum toroidal algebra

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.

Differential-difference system related to toroidal Lie algebra

We present a novel differential-difference system in (2+1)-dimensional space-time (one discrete, two continuum), arisen from the Bogoyavlensky's (2+1)-dimensional KdV hierarchy. Our method is based on the bilinear identity of the hierarchy, which is related to the vertex operator representation of the toroidal Lie algebra $\sl_2^{tor}$.

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

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…

仿射杨代数作为量子扭代数的极限

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。

Quantum toroidal algebras and motivic Hall algebras I. Hall algebras for singular elliptic curves

We consider the motivic Hall algebra of coherent sheaves over an irreducible reduced projective curve of arithmetic genus $1$. We introduce the composition subalgebra in the singular curve case, and show that it is isomorphic to the composition subalgebra for a smooth elliptic curve. As in the case of smooth elliptic case studied by Burban and Schiffmann, the reduced Drinfeld double of the composition subalgebra is isomorphic to the quantum toroidal algebra for $\mathfrak{gl}_1$ (also called Ding-Iohara-Miki algebra), and it inherits automorphisms induced from equivalences of the associated de…

Quantum Toroidal Comodule Algebra of Type $A_{n-1}$ and Integrals of Motion

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.

两种洗牌代数的故事

A tale of two shuffle algebras

本文从量子仿射群赋值表示与R矩阵出发，给出量子环面代数“上半”“下半”部分的洗牌代数表示，得到可延拓水平量子仿射子代数Drinfeld-Jimbo余积的新拓扑余积。

Little Strings, Quasi-topological Sigma Model on Loop Group, and Toroidal Lie Algebras

We study the ground states and left-excited states of the A_{k-1} N=(2,0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.

Evaluation modules for quantum toroidal ${\mathfrak{gl}}_n$ algebras

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…

Quantum toroidal $\mathfrak{gl}_1$ algebra : plane partitions

In third paper of the series we construct a large family of representations of the quantum toroidal $\gl_1$ algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight $\gl_\infty$-modules.

Computing the R-matrix of the quantum toroidal algebra

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…

The $(\mathfrak{gl}_m,\mathfrak{gl}_n$) duality in the quantum toroidal setting

On a Fock space constructed from $mn$ free bosons and lattice ${\Bbb {Z}}^{mn}$, we give a level $n$ action of the quantum toroidal algebra $\mathscr {E}_m$ associated to $\mathfrak{gl}_m$, together with a level $m$ action of the quantum toroidal algebra ${\mathscr E}_n$ associated to ${\mathfrak {gl}}_n$. We prove that the $\mathscr {E}_m$ transfer matrices commute with the $\mathscr {E}_n$ transfer matrices after an appropriate identification of parameters.

Hierarchy of (2+1)-dimensional nonlinear Schroedinger equation, self-dual Yang-Mills equation, and toroidal Lie algebras

The hierarchy structure associated with a (2+1)-dimensional Nonlinear Schroedinger equation is discussed as an extension of the theory of the KP hierarchy. Several methods to construct special solutions are given. The relation between the hierarchy and a representation of toroidal Lie algebras are established by using the language of free fermions. A relation to the self-dual Yang-Mills equation is also discussed.

Integrals of motion from quantum toroidal algebras

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors. That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model. We also discuss the (gl(m),gl(n)) duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ans…

洗牌代数、格路与量子扭椭圆$\mathfrak{gl}_{n|m}$

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。