Vertex representations for twisted affine Lie algebra of type A_{2l}^{(2)}
Limeng Xia, Naihong Hu, Xiaotang Bai
本文构造了A_{2l} ^{（2）}型扭仿射李代数的一个不可约顶点模.
In this paper, we construct an irreducible vertex module for twisted affine Lie algebra of type A_{2l}^{(2)}.
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Vogan diagrams of affine twisted Lie superalgebras
Vogan图是基于极大紧Cartan子代数的扭曲仿射超代数的Cartan对合的Dynkin图。本文构造了扭仿射超代数的Vogan图。这是完成vogan图到超代数分类的一部分。
Denominator Identity for Twisted Affine Lie Superalgebras
李超代数分母恒等式的研究是最近由M. Gorelik，V.G. Kac，P.Moseneder Frajria，I. Musson，P. Papi，M.胁本和作者。本文将这些恒等式推广到扭仿射情形，将结果推广到所有仿射李超代数。
The Drinfled realization of quantum affine algebras has been tremendously useful since its discovery. Combining techniques of Beck and Nakajima with our previous approach, we give a complete and conceptual proof of the Drinfeld realization for the twisted quantum affine algebras using Lusztig's twisted braid group action.
The explicit Verlinde formula for the dimension of conformal blocks, attached to a marked projective curve $Σ$, a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and integrable highest weight modules of a fixed central charge of the corresponding affine Lie algebra $\hat{L}(\mathfrak{g})$ attached to the marked points, requires (among several other important ingredients) a Lie algebra cohomology vanishing result due to C. Teleman for the positive part $\hat{L}^+(\mathfrak{g})$ with coefficients in the tensor product of an integrable highest weight module with copies of finite dimensional e…
The work is devoted to a probably new connection between deformed Virasoro algebra and quantum $\widehat{\mathfrak{sl}}_2$. We give an explicit realization of Virasoro current via vertex operators of level 1 integrable representation of $\widehat{\mathfrak{sl}}_2$. The same is done for a twisted version of deformed Virasoro algebra.
The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of twisted affine Lie algebras. Conjectures for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, usin…
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本研究利用高斯分解，建立了AI型对称对对应的扭曲$q$-Yangian与仿射$\imath$量子群的显式同构，并由此得到该类仿射$\imath$量子群的PBW型基。
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我们将I.将非扭仿射李代数的Frenkel轨道理论推广到扭仿射李代数的情形，并利用了某些非连通紧李群的特征标公式。
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本文通过仿射 ı 量子群 Drinfeld 呈现的退化构造了分裂型扭 Yangian，证明其具有 PBW 型基且为扭流代数的形变，并猜想其与 RTT 呈现中的对应代数同构。
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本文将Zhu关于整数共形权分次顶点算子代数上模的特征标的定理推广到有理共形权分次顶点算子超代数的情形.为了恢复特征标的SL_2（Z）-不变性，有必要考虑扭曲模和普通模。它也被证明是必要的，在描述空间的共形块在超对称的情况下，包括某些“奇迹”的模块旁边的痕迹和supertrace。我们证明了这样补充后的超迹函数集是有限维SL_2（Z）-不变空间。我们用几个例子来结束这篇论文。
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
This is a continuation of our work to understand vertex operator algebras using the geometric properties of varieties attached to vertex operator algebras. For a class of vertex operator algebras including affine vertex operator algebras associated to a finite dimensional simple Lie algebra $\mathfrak{g}$, we describe their varieties of semi-conformal vectors by some matrix equations. These matrix equations are too complicated to be solved for us. However, for affine vertex operator algebras associated to the simple Lie algebra $\mathfrak{g}$, we find the adjoint group $G$ of $\mathfrak{g}$ ac…
Boson-fermion correspondence of type B and twisted vertex algebras
类型A的玻色子-费米子对应是两个超顶点代数之间的同构（因此只有在$z = w$上的算子乘积展开式才有奇点）。B型玻色子-费米子对应在许多领域中起着同样重要的作用，包括表示论、可积系统、随机矩阵理论和随机过程。但是描述它的顶点算子在$z = w$和$z = -w$上的算子乘积展开式都有奇点，因此需要一个比超顶点代数更一般的概念。本文提出了这样一个概念：扭顶点代数的概念，它推广了超顶点代数的概念。B型对应的两边构成扭顶点代数的两个例子。因此，B型玻色子-费米子对应是两个扭曲顶点代数之间的同构。
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.
Twisted global section functor for D-modules on affine Grassmannian
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On a twisted Euler-Poincaré pairing for graded affine Hecke algebras
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Construction of perfect crystals conjecturally corresponding to Kirillov-Reshetikhin modules over twisted quantum affine algebras
假设单缀量子仿射代数上Kirillov-Reshetikhin模的完美晶体基的存在性，构造了扭曲量子仿射代数的完美晶体，并证明了所构造的晶体与扭曲量子仿射代数上Kirillov-Reshetikhin模的完美晶体基同构.
We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.
We establish the $Q \widetilde{Q}$-systems for the twisted quantum affine algebras that were conjectured in arXiv:1606.05301. We develop the representation theory of Borel subalgebra of twisted quantum affine algebras and we construct their prefundamental representations. We also propose a general conjecture on the relations between twisted and non-twisted types. We prove this conjecture for some particular classes of representations, including prefundamental representations.
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我们研究Lusztig的t-模拟的重量多重性与扭曲仿射Kac-Moody代数的一级表示。利用Macdonald和Cherednik的常数项恒等式，得到了相应的t-弦函数的显式封闭形式表达式.封闭形式涉及扭曲仿射代数的分次片的广义指数，被认为是潜在的有限维简单李代数的模块。这扩展了以前的工作水平1 t-弦函数的无扭简单花边仿射Kac-Moody代数。
$N$-point locality for vertex operators: normal ordered products, operator product expansions, twisted vertex algebras
本文研究满足$N$ -点局部性的域及其性质。我们得到了$N$ -点局部场用δ函数的导数和Bell多项式表示的留数公式。本文引入$N$ -点局部域的后代空间的概念，它包括正规序积和算子乘积展开的系数。我们证明了$N$ -点局域场的例子包括产生B、C和D-A型玻色子-费米子对应的顶点算符。我们将这些顶点算子的正规序积应用到双无限秩李代数$b_{\infty}, c_{\infty}, d_{\infty}$的表示论中。最后，我们证明了由$N$ -点局部场及其后代生成的场论具有扭曲顶点代数的结构。
In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer in J. Algebra, 163, 861--879 (1994). We also show that for a non-finite and non-cofinite biclosed set $B$ in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed $B$-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Final…
We show that sheet closures appear as associated varieties of affine vertex algebras. Further, we give new examples of non-admissible affine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures from our previous paper and give new examples of lisse affine W-algebras.
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.
In this paper we classify the irreducible integrable modules for the twisted toroidal extended affine Lie algebras with center acting non-trivially.
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.
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本文对一般扭曲型特殊仿射群概形证明 level one 扭曲仿射 Demazure 模与环面不动点子概形函数环之间的对偶定理，并由此确定任意仿射 Schubert 簇的光滑轨迹，证实 Haines-Richarz 猜想。
The Vertex Algebra $M(1)^+$ and Certain Affine Vertex Algebras of Level -1
给出了中心荷为$\ell$的顶点算子代数$M(1)^+$的陪集实现.我们将$M(1)^+$实现为顶点代数$L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}Λ_0) \otimes L_{C_{\ell} ^{(1)}}(-\tfrac{1}{2}Λ_0)$中的某些水平-1的仿射顶点代数的交换子。我们证明了简单顶点代数$L_{C_{\ell} ^{(1)}}(-Λ_0)$可以共形地嵌入到$L_{A_{2 \ell -1} ^{(1)}} (-Λ_0)$中并找到相应的分解.我们还研究了$L_{C_{\ell} ^{(1)}}(-Λ_0)$中的某些陪集子代数。
We apply the theory of $ϕ$-coordinated modules, developed by H.-S. Li, to the Etingof--Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. We prove, for a certain associate $ϕ$ of the one-dimensional additive formal group, that any $ϕ$-coordinated module for the level $c\in\mathbb{C}$ quantum affine vertex algebra is naturally equipped with a structure of restricted level $c$ module for the quantum affine algebra in type $A$ and vice versa. Moreover, we show that any $ϕ$-coordinated module is irreducible with respect to the action of the quantum affi…
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.
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
玻色-费米子对应是数学物理中几个领域交叉的重要现象：表示论、顶点代数与共形场论、可积系统、数论、上同调。两个这样的对应是众所周知的：类型A和B（以及它们的超扩展）。作为本文的主要结果，我们提出了一个新的D-A型玻色子-费米子对应。进一步定义了$N$阶扭顶点代数的概念，它是超顶点代数的推广.我们开发的bicharacter建设，我们用来构建类的例子扭曲顶点代数，以及推导公式的算子乘积展开（OPEs），解析连续和正常有序产品。通过使用底层的Hopf代数结构，我们证明了一般的双特征公式的真空期望值的三组重要的例子。我们证明了B、C和D-A型对应是扭顶点代数的同构。
Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group
该研究证明仿射旗簇上的扭曲惠特克范畴与混合量子群表示范畴等价，且一般情形下量子范畴O和仿射旗上的扭曲惠特克D-模范畴等价，丰富了量子群与几何表示论的联系。
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.
Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting
本文将Kostant等人关于有限维李代数的多重表示和三次Dirac算子的经典结果推广到仿射李代数和扭仿射三次Dirac算子的情形。我们证明了在这种情况下的一个类似的沃根猜想的无穷小字符的哈里什-钱德拉模块的狄拉克上同调。对于我们的计算，我们使用机器的李共形和顶点代数。
We study embeddings of the simple admissible affine vertex algebras $V_k(sl(2))$ and $V_k(osp(1,2))$, $k \notin {\Bbb Z}_{\ge 0}$, into the tensor product of rational Virasoro and $N=1$ Neveu-Schwarz vertex algebra with lattice vertex algebras. We prove that the admissible affine vertex algebra $V_k(sl(2))$ can be embedded into vertex algebra $L^{Vir} (c_{p,p'}, 0) \otimes Π(0)$ where $L^{Vir} (c_{p,p'}, 0) $ is suitable minimal Virasoro vertex algebra and $Π(0)$ is a vertex algebra of lattice type. By using these realizations we construct a family of weight, logarithmic and Whittaker $\wideha…
We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.
On certain categories of modules for twisted affine Lie algebras
我们将可积不可约$\hat{g}[σ]$ -模分类到范畴E和C中，证明了E包含著名的赋值模，C统一了最高权模、赋值模及其张量积模.
Twisted vertex operators and Bernoulli polynomials
利用顶点算子代数及其扭曲模理论中的一般原理，得到了圆上微分算子李代数的中心扩张对任意扭曲自同构的玻色扭曲结构.该构造涉及到基本的伯努利多项式。我们开发了新的身份和原则的理论顶点算子代数和扭曲的模块，并解释建设应用一般的结果，包括身份，我们称之为“修改弱结合”，海森堡顶点算子代数。本文给出了证明和进一步的解释先前公布的结果。它是对第二作者不久前发表的工作的扭曲顶点算子的推广，并作为特例包括了该工作的主要结果的证明。
We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces.
Some general results on conformal embeddings of affine vertex operator algebras
给出了任意层上仿射李代数与顶点算子代数共形嵌入的一般判据。利用这个准则，我们构造了新的共形嵌入在容许有理和负整数水平。特别地，我们构造了简单李代数的Dynkin图的自同构的所有剩余的共形嵌入。利用顶点算子代数融合规则的概念，得到了相应分解的半单性。
We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We then give a criterion for these local systems, in terms of their parameters, to extend cleanly in the sense of $\mathcal{D}$-modules.
Twisted Vertex Operators and A-D-E Representations
本文研究了偶整${\mathbf Z}_2$ -格上的扭顶点算子代数，给出了一类$A$，$D$，$E$型仿射李代数的基本表示及其不可约分解的系统构造.
In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.
Weyl modules for the twisted loop algebras
Weyl模的概念，以前定义的无扭仿射代数，这里扩展到扭曲仿射代数。我们描述了一个识别的Weyl模的扭曲仿射代数与适当选择的Weyl模的无扭曲仿射代数。这一识别使我们能够使用已知的结果在untwisted的情况下，计算的维数和字符的Weyl模的扭曲代数。
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。
Let $Z \subset \mathbb{A}^k$ be an affine scheme over $\C$ and $\J Z$ its jet scheme. It is well-known that $\mathbb{C}[\J Z]$, the coordinate ring of $\J Z$, has the structure of a commutative vertex algebra. This paper develops the orbifold theory for $\mathbb{C}[\J Z]$. A finite-order linear automorphism $g$ of $Z$ acts by vertex algebra automorphisms on $\mathbb{C}[\J Z]$. We show that $\mathbb{C}[\J^g Z]$, where $\J^g Z$ is the scheme of $g$--twisted jets has the structure of a $g$-twisted $\mathbb{C}[\J Z]$ module. We consider spaces of orbifold coinvariants valued in the modules $\mathb…
Twisted loop groups and their affine flag varieties
本文发展了Laurent幂级数局部域k（（t））上约化群的仿射旗簇及其Schubert簇的理论，其中k是完满域.这可以被看作是一个推广的理论，仿射旗品种循环组的“扭曲的情况下”，我们的结果的一个后果是，我们的建设也包括旗品种的卡茨-穆迪李代数的仿射型。给出了k（（t））上固定群上的部分旗簇上的自然充裕线丛的整体瓣空间的维数的一个凝聚猜想，以及在Shimura簇的局部模型中的一些应用.
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.
We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.
We analyze the asymptotic behavior of certain twisted orbital integrals arising from the study of affine Deligne-Lusztig varieties. The main tools include the Base Change Fundamental Lemma and $q$-analogues of the Kostant partition functions. As an application we prove a conjecture of Miaofen Chen and Xinwen Zhu, relating the set of irreducible components of an affine Deligne-Lusztig variety modulo the action of the $σ$-centralizer group to the Mirkovic-Vilonen basis of a certain weight space of a representation of the Langlands dual group.
Twisted vertex representations and spin characters
我们建立了一个新的群论实现的基本表示的扭曲仿射和扭曲环面代数的ADE类型在相同的精神，我们的新方法的麦凯对应。我们的顶点算子构造为扭曲超八面体群与任意有限群的圈积的自旋覆盖的特征标表提供了统一的描述。
Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(kΛ_0)$ called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor product of the twisted module $V_L^T$ for the lattice vertex operator algebra $V_L$. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in arXiv:math/0102113.
GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian
本文构造了类型AI扭曲Yangian的GKLO表示，定义其截断，并证明截断的扭曲Yangian量子化仿射Grassmannian横截切片商的概形。
Algebras of twisted chiral differential operators and affine localization of $\frak{g}$-modules
提出了第一Pontrjagin类为零的代数流形上的手征微分算子代数的概念。我们发现，这样的代数具有家庭的模块取决于无穷多个复杂的参数，我们分类在相应的代数扭曲微分算子。如果基础流形是旗流形，我们的构造恢复模的仿射李代数参数化的opers在朗兰兹对偶李代数。空间的整体瓣的“最小”这样的模块是不可约的$\ghat$ -模块和所有不可约的$\frak{g}$ -可积的$\ghat$ -模块在关键的水平出现在这种方式。
Twisted $Γ$-Lie algebras and their vertex operator representations
设$Γ$是非零复数乘法群$\mathbb{C}^*$的一般子群。定义了一类与$Γ$相关的李代数，称为扭$Γ$ -李代数，它是扭仿射李代数的自然推广.从$\mathbb Z^N$的任意偶子格$Q$和$\mathbb Z^N$的任意有限阶等距保$Q$出发，构造了一类作用于广义Fock空间的扭曲$Γ$ -顶点算子，它们为某些扭曲$Γ$ -李代数提供了不可约表示.作为应用，这恢复了许多已知的无穷维李代数的顶点算子实现，如扭曲仿射李代数，$A$型的扩展仿射李代数，$A$和$B$系列的三角李代数，酉李代数，$BC$分次李代数.
We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove duality, weak associativity, a Jacobi identity, a generalized commutator formula, generalized weak commutativity, and convergence and commutativity for products of more than two operators involving twist vertex operators. These properties of twist vertex operators play an important role in the author's recent general, direct and explicit construction of (lower-bounded generalized) twisted modules.
Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras
本文通过引入无限长Lakshmibai-Seshadri路径，建立了扭曲量子齐性代数零级极值权重模块与晶体基之间的同构关系。
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。
Twisted representations of vertex operator superalgebras
本文对顶点算子超代数V和有穷级的自同构g给出了A_g（V）理论的一个类似.建立了g-扭V-模与A_g（V）-模之间的关系。证明了若V是g-有理数，则A_g（V）是有限维半单结合代数，且只有1000个不可约g-扭V-模.
The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an $SL(2,\mathbb{Z})$-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram autom…
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