Johan van de Leur
Using $n$ finite order automorphisms on a simple complex Lie algebra we construct twisted $n$-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case $n=2$ we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Saito.
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The paper is to classify irreducible integrable modules for the twisted full toroidal Lie algebra with some technical conditions. The twisted full toroidal Lie algebra are extensions of multiloop algebra twisted by sevaral finite order automorphisms. The result genaralizes a result by Fu Jiayuan and Cuipo Jiang, where they consider only one automorphism.
Toroidal Lie algebras are generalizations of affine Lie algebras. In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras. In this paper we give a presentation for the twisted toroidal Lie algebras of type $A$ and $D$ constructed by Fu and Jiang.
Using free fields, we construct a bosonic realization of toroidal Lie algebras of type $A_{2n-1}, A_{2n}, D_{n+1}, D_4$ which are twisted by a Dynkin diagram automorphism. This realization is based upon the recently found Moody-Rao-Yokonuma-like presentation. In particular, our construction contains the first bosonic realization of the twisted affine algebra $D_4^{(3)}$.
In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus $LT$ which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial $LT$-action and where the centre of the underlying Lie algebra acts trivially.
In this paper, we construct a fermionic realization of the twisted toroidal Lie algebra of type $A_{2n-1}, D_{n+1}, A_{2n}$ and $D_4$ based on the newly found Moody-Rao-Yokonuma-like presentation.
In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with center acting non-trivially.
Weyl modules for twisted toroidal Lie algebras
本文扩展了扭曲环面李代数 $\mathcal{T}(μ)$ 的Weyl模概念，证明其第一层全局Weyl模同构于扭曲仿射李代数第一层表示与格顶点代数的张量积，并计算了第一层局部Weyl模的分次特征标。
The Toroidal Lie algebras are n variable genaralizations of Affine Kac-Moody Lie algebras. As in the affine Lie algebras there exists finite order auto= morphisms corresponding to Dynkin diagram automorphisms. The fixed point sub= algebras are called twisted toroidal Lie algebras. In this paper we construt faithfull representations for the twisted toroidal Lie algebras (this includes the non-twisted case also) useing methods developed by Lepowsky-Wilson [LW]. This construction recovers the result by Eswara Rao-Moody [EM] in the homogeneous picture and by Yuly Billig [B1] in the principal pictu…
In this paper, we study modules for the Twisted Full Toroidal Lie Algebras (TFTLA). We define a category of bounded modules for TFTLA and classify all the irreducible modules in that category. A class of irreducible bounded modules are explicitly constructed by Michael Lau and Yuly Billig using Vertex operator algebras. We show that they exhaust all the irreducible bounded modules. Our plan is to describe the top spaces of the bounded irreducible modules for TFTLA and prove they in fact coincide with the top spaces of the modules constructed by Lau and Billig. When the top spaces become isomor…
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.
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 $\fg$ be an affine Kac-Moody algebra, and $μ$ a diagram automorphism of $\fg$. In this paper, we give an explicit realization for the universal central extension $\wh\fg[μ]$ of the twisted loop algebra of $\fg$ related to $μ$, which provides a Moody-Rao-Yokonuma presentation for the algebra $\wh\fg[μ]$ when $μ$ is non-transitive, and the presentation is indeed related to the quantization of toroidal Lie algebras.
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.
This is an expository article on representation theory of toroidal Lie algebras. We summerize all the results on representation theory of toroidal Lie algebras obtained in the last fifteen years. Apart from that a natural genaralization of Virasoro Lie algebra is introduced and the related results are stated. A genaralization of the semi direct product of affine Lie algebra and Virasoro Lie algebra is introduced which we call the full toroidal Lie algebra. The connection to the theory of Extended Affine Lie Algebras is explained.
In this paper we construct a large class of modules for toroidal Lie superalgebras. Toroidal Lie superalgebras are universal central extension of G tensor A where G is a basic classical Lie superalgebra and A is a Laurent polynomial ring in sevaral variables. The case where G is simple finite dimensional Lie algebra is included.
We study irreducible modules for Toroidal Lie-algebras with finite dimensional weight spaces. First note that Toroidal Lie-algebras have infinite dimensional center. In genaral the infinite dimensional center does not act as scalars on an irreducible module but acts as invertible central operators. We prove in this paper the study of irreducible modules for Toroidal Lie-algebras with finite dimensional weight spaces is reduced to the study of irreducible modules where the center acts as scalars.
Toroidal Lie algebras are universal central extentions of the finite dimensional simple Lie algbera tensored with Laurent Polynomials in several commuteing variables. In this paper we classify irreducible integrable modules for Toroidal Lie algebras with finite dimensional wieght spaces. They are tensor product of evaluation modules for the finite dimensional Lie algebra when the center acts by zero. In the case when the center acts nontrivially they are tensor product of evalution representations of the underlying affine Lie algebra twisted by an automorphism of the Toroidal Lie algebras. The…
In this paper we classify irreducible integrable representations of loop toroidal Lie algebras with finite dimensional weight spaces. In both the cases we classify modules, when a part of center acts non-trivially and trivially on modules.
Integrable Modules of Map full Toroidal Lie Algebras
本文针对具有有限维权空间的Map全环面李代数，研究其可积表示范畴Cf中的不可约对象，证明这类对象均为单点赋值模，即底层全环面李代数的不可约可积模。
In this paper we study the subcategory of finite-length objects of the category of positive level integrable representations of a toroidal Lie algebra. The main goal is to characterize the blocks of the category. In the cases when the underlying finite type Lie algebra associated with the toroidal Lie algebra is simply-laced, we are able to give a parametrization for the blocks.
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 $\widetilde{\mathfrak g}$.
Classification of irreducible Harish-Chandra modules over map full toroidal Lie algebras
本文分类了映射全环面李代数上的不可约Harish-Chandra模，证明其必为尖点模或最高权模，且均为单点赋值模。
The irreducible integrable representations with finite-dimensional weight spaces of toroidal Lie algebras on which the center acts non-trivially were classified by S.Eswara Rao. In this paper we give a compact proof of the results that lead to the classification theorem. Further we establish a necessary and sufficient condition under which two such irreducible modules are isomorphic.
We show that the representation theory for the toroidal extended affine Lie algebras is controlled by a vertex operator algebra which is a tensor product of four VOAs: a sub-VOA of the hyperbolic lattice VOA, two affine VOAs and a Virasoro VOA. A tensor product of irreducible modules for these VOAs admits the structure of an irreducible module for the toroidal extended affine Lie algebra. We also show that for N=12, the sub-VOA of the hyperbolic lattice VOA becomes an exceptional irreducible module for the extended affine Lie algebra of rank 0.
In this paper we study Weyl modules for a toroidal Lie algebra $\CT$ with arbitrary $n$ variables. Using the work of Rao \cite{1995}, we prove that the level one global Weyl modules of $\CT$ are isomorphic to suitable submodules of a Fock space representation of $\CT$ upto a twist. As an application, we compute the graded character of the level one local Weyl module of $\CT$, thereby generalising the work of Kodera \cite{ko}.
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $τ$. More specifically, we define and study two categories $\mathcal{E}_τ$ and $\mathcal{C}_τ$ of $τ$-modules using generating functions, where $\mathcal{E}_τ$ is proved to contain the evaluation modules while $\mathcal{C}_τ$ contains certain restricted $τ$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $\mathcal{E}_τ$ and $\mathcal{C}_τ$.
Let $\fg$ be any untwisted affine Kac-Moody algebra, $μ$ any fixed complex number, and $\wt\fg(μ)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bmλ=(λ_1, \cdots, λ_k)$ of weights of $\fg$, and $k$-tuple $\bm{a}=(a_1,\cdots, a_k)$ of distinct non-zero complex numbers, we construct a class of modules $\wt V(\bmλ,\bm{a})$ for the extended affine Lie algebra $\wt\fg(μ)$. We prove that the $\wt\fg(μ)$-module $\wt V(\bmλ,\bm{a})$ is completely reducible. We also prove that the $\wt\fg(μ)$-module $\wt V(\bmλ,\bm{a})$ is integrable when all weights $λ_i$ i…
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 introduce a category B of bounded modules for the toroidal Lie algebras and study irreducible modules in B. We show that one of the irreducible modules in this category, L(T_0), admits a structure of a vertex operator algebra. We prove that L(T_0) factors into a tensor product of a sub-VOA of a hyperbolic lattice VOA and a simple VOA associated with a twisted Virasoro-affine Lie algebra. Every irreducible module in category B is a VOA module for a slightly larger VOA V(T_0). Knowing the structure of V(T_0), we are able to give explicit realizations for all irreducible modules in category B …
We generalize I. Frenkel's orbital theory for non twisted affine Lie algebras to the case of twisted affine Lie algebras using a character formula for certain non-connected compact Lie groups.
Integrable Representations for Toroidal Lie Algebras Co-ordinated by Rational Quantum Torus
本文研究了由有理量子扭圈协调的扭形李代数的不可约可积表示，完成其在中心作用平凡情况下的分类。
Let $Γ$ be a generic subgroup of the multiplicative group $\mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Γ$, called twisted $Γ$-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $\mathbb Z^N$ and an arbitrary finite order isometry of $\mathbb Z^N$ preserving $Q$, we construct a family of twisted $Γ$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Γ$-Lie algebras. As application, this recovers a number o…
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.
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.
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.
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…
For a twisted affine Lie superalgebra with nonzero odd part, we study {tight irreducible weight modules} with bounded weight multiplicities and show that if the action of nonzero real vectors of each affine component of the zero part is neither completely injective nor completely locally nilpotent, then these modules are parabolically induced.
We discuss the category $\cal I$ of level zero integrable representations of loop algebras and their generalizations. The category is not semisimple and so one is interested in its homological properties. We begin by looking at some approaches which are used in the study of other well--known non--semisimple categories in the representation theory of Lie algebras. This is done with a view to seeing if and how far these approaches can be made to work for $\cal I$. In the later sections we focus first on understanding the irreducible level zero modules and later on certain universal modules, the …
Let A_k denote the twisted group algebra of the symmetric group S_k, whose representations correspond to the nonlinear projective representations of S_k. We establish a duality relation between A_k and a Lie superalgebra q(n), sometimes called the ``queer'' Lie superalgebra. This duality clarifies Schur's formula, in the sense that the Schur-Weyl duality clarifies Frobenius' formula.
In this paper, we characterize quasi-integrable modules, of nonzero level, over twisted affine Lie superalgebras. We show that quasi-integrable modules are not necessarily highest weight modules. We prove that each quasi-integrable module is parabolically induced from a cuspidal module, over a finite dimensional Lie superalgebra having a Cartan subalgebra whose corresponding root system just contain real roots; in particular, the classification of quasi-integrable modules is reduced to the known classification of cuspidal modules over such Lie superalgebras.
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 $c_2=0$.
Simple Modules For Twisted Hamiltonian Extended Affine Lie Algebras
本文对扭哈密顿扩展仿射李代数在有限维中心非平凡作用下的不可约可积模进行分类，并利用其不同于扭全环面李代数的三角分解详细刻画了最高权空间。
This work provides the first step toward the classification of irreducible finite weight modules over twisted affine Lie superalgebras. We study all such modules whether the canonical central element acts as a nonzero multiple of the identity map or not. Moreover, we reduce the classification of some subclasses of irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal superalgebras which is known by a work of Dimitrov, Mathieu and Penkov.
Classification of irreducible Harish-Chandra modules over full toroidal Lie algebras and higher-dimensional Virasoro algebras
本文对满Toroidal李代数和高维Virasoro代数上的不可约Harish-Chandra模进行了分类，揭示了其结构特征。
Let $G_\cpl$ be a connected complex reductive Lie group, and $G$ be a real form. Let $(π,V)$ be a finite-dimensional irreducible representation of $G$. Assume $(π,V)$ admits a $G$ invariant hermitian form. {In \cite{Signature-of-a-rep-of-reductive}, an analog of the Weyl dimension formula that instead computes the signature of the invariant form for finite dimensional representations of complex Lie groups is given. The goal of this paper is to give a short alternate proof as an application of a more general twisted character theory of \cite{Dirac-Index-and-Twisted-Characters}.
Schur-Weyl Duality for Toroidal Algebras of Type $A$
该研究提出并证明了A型经典2-环面李代数商代数的Schur-Weyl对偶类似物，还给出了将其推广到m>2的m-环面情形的方法，拓展了Schur-Weyl对偶的适用范围。
Zero-level integrable modules over twisted affine Lie superalgebras
本文研究了扭曲仿射李超代数上的零级可积有限重量模的特征化，通过诱导方法和二次李超代数的结构分析，给出了相关模的完整分类。
The Whittaker module $M_ψ$ and its quotient Whittaker module $L_{ψ, ξ}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[τ]$ are studied. For nonsingular type, it is proved that if $ξ\neq 0$, then $L_{ψ,ξ}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[τ]$-module of type $ψ$ with ${\bf k}$ acting as a non-zero scalar $ξ$ is isomorphic to $L_{ψ,ξ}$. Furthermore, for $ξ=0$, all Whittaker vectors of $L_{ψ, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{ψ, ξ}$ with $ξ\neq 0$ are fully characterized.
Derivations of Twisted Loop Algebras of Minimal Q-graded Subalgebras
本文研究了极小Q-分级子代数的扭曲环形代数的导出，推导出导出代数的分解公式，并确定了同质几乎内导出和扭曲affinization。
The purpose of this paper is to introduce twisted $\mathcal{O}$-operators on $3$-Lie algebras. We define a cohomology of a twisted $\mathcal{O}$-operator $T$ as the Chevalley-Eilenberg cohomology of a certain $3$-Lie algebra induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted $\mathcal{O}$-operators from cohomological points of view. Furthermore, we introduce and study $3$-NS-Lie-algebras as the underlying structure of twisted $\mathcal{O}$-operators on $3$-Lie algebras. Finally, we investigate twisted $\mathcal{O}$-o…
We construct irreducible modules for twisted toroidal Lie algebras and extended affine Lie algebras. This is done by combining the representation theory of untwisted toroidal algebras with the technique of thin coverings of modules. We illustrate our method with examples of extended affine Lie algebras of Clifford type.
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 $\sl_2^{tor}$.
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。
An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
Folding shuffle algebras and twisted $q$-characters
通过引入折叠洗牌代数，证明了量子非扭曲仿射代数模的某些 $q$-特征与其扭曲对应物相等的Hernandez猜想，并将结果推广到具有自同构的任意箭图，特别地定义并描述了扭曲量子环面代数。
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.
In this paper, we prove the categories of lower bounded twisted modules of positive integer levels for simple vertex operator algebras associated with affine Lie algebras and general automorphisms are semisimple, using the twisted generalization of Zhu's algebra for these vertex operator algebras, constructed in \cite{HY}. We also show that the category of lower bounded twisted modules for a general automorphism is equivalent to the category of lower bounded twisted modules for the corresponding diagram automorphism.
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.
In this paper, we construct an irreducible vertex module for twisted affine Lie algebra of type A_{2l}^{(2)}.
Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $kΛ_0$ and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in arXiv:math/0102113.
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