Luan Pereira Bezerra, Lucas Calixto, Vyacheslav Futorny, Iryna Kashuba
We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra of type A.
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On integral forms for vertex superalgebras associated with affine Lie superalgebras and their modules
本文研究仿射顶点超代数及其模的积分形式，通过构造仿射李超代数通用包络超代数的积分形式，并利用Chevalley基推广Kostant-Garland积分形式理论，进而构造基于仿射李超代数的顶点（算子）超代数及其模的积分形式，并探讨积分形式包含共形向量以及反梯度模中的积分形式。
Let Uq(g) be the quantum affine superalgebra associated with an affine Kac-Moody superalgebra g which belongs to the three series osp(1|2n)^(1),sl(1|2n)^(2) and osp(2|2n)^(2). We develop vertex operator constructions for the level 1 irreducible integrable highest weight representations and classify the finite dimensional irreducible representations of Uq(g). This makes essential use of the Drinfeld realisation for Uq(g), and quantum correspondences between affine Kac-Moody superalgebras, developed in earlier papers.
Orbifold theory of the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$
本文开展仿射顶点算子超代数的轨形理论研究，完成单仿射顶点算子超代数 $L_{\widehat{osp(1|2)}}(k,0)$ 轨形的不可约模分类并确定其融合规则。
We study the quantum affine superalgebra $U_q(Lsl(M,N))$ and its finite-dimensional representations. We prove a triangular decomposition and establish a system of Poincaré-Birkhoff-Witt generators for this superalgebra, both in terms of Drinfel'd currents. We define the Weyl modules in the spirit of Chari-Pressley and prove that these Weyl modules are always finite-dimensional and non-zero. In consequence, we obtain a highest weight classification of finite-dimensional simple representations when $M \neq N$. Some concrete simple representations are constructed via evaluation morphisms.
We study Duflo-Serganova functor for non-twisted affine Lie superalgebras and affine vertex superalgebras.
Tensor product degenerate spin affine Hecke superalgebras
本文定义了高阶退化自旋仿射Hecke超代数，建立其与高阶退化仿射Hecke-Clifford超代数的同构关系，推广了退化情形下两类超代数的对应关系，丰富了超代数结构理论研究。
This article classifies the Vogan diagram of the affine untwisted Kac Moody superalgebras.
We prove a denominator identity for non-twisted affine Lie superalgebras with zero dual Coxeter number.
Generalized root systems (GRS), that were introduced by V. Serganova, are a generalization of finite root systems (RS). We define a generalization of affine root systems (ARS), which we call $\textit{affine generalized root systems}$ (AGRS). The set of real roots of almost every symmetrizable affine indecomposable Kac-Moody superalgebra is an irreducible AGRS. In this paper we classify all AGRSs and show that almost every irreducible AGRS is the set of real roots of a symmetrizable affine indecomposable Kac-Moody superalgebra.
Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences
本文基于RTT表示研究了任意01-序列下量子仿射一般线性超代数的有限维不可约表示，构造了相容的PBW基，确定了有限维性充要条件并推广至仿射情形，为量子仿射超代数表示研究奠定基础。
We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.
Drinfeld Realization for Quantum Affine Orthosymplectic Superalgebras
本文通过给出任意奇偶性下辫子算子的显式公式，解决了量子仿射正交辛超代数中辫子群作用定义不完整的问题，并利用该作用引入了新的Drinfeld实现，证明了Drinfeld-Jimbo量子群是其满同态像。
Conjugation Theorem For Affine Kac-Moody Superalgebras
本文将Cartan子代数的共轭定理从对称可约Kac-Moody代数推广到仿射Kac-Moody超代数，并深入探讨了其结构及虚根空间的关系。
Ordinary modules for affine vertex operator superalgebras
本文证明边界容许水平下仿射顶点算子超代数的普通模范畴有限半单，等同于有限长广义模范畴，并进一步证明其为ribbon融合超范畴。
Admissible modules over affine Lie superalgebras: The final step in the characterization
本文完成对仿射李超代数上简单可接受模的特征化，解决了零级模的相关问题。
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 study a certain family of finite-dimensional simple representations over quantum affine superalgebras associated to general linear Lie superalgebras, the so-called fundamental representations: the denominators of rational $R$-matrices between two fundamental representations are computed; a cyclicity (and so simplicity) condition on tensor products of fundamental representations is proved.
Rao and Zhao classified the irreducible integrable modules with finite dimensional weight spaces for the untwisted affine superalgebras which are not $\hat{A}(m,n)$ ($m\ne n$) or $\hat{C}(m)$. Here we treat the latter affine superalgebras to complete the classification. The problem boils down to classifying the irreducible zero-level integrable modules with finite dimensional weight spaces for these affine superalgebras, which is solved in this paper. We note in particular that such modules for $\hat{A}(m,n)$ ($m\ne n$) and $\hat{C}(m)$ must be of highest weight type, but are not necessarily l…
There is a surprising isomorphism between the quantised universal enveloping algebras of osp(1|2n) and so(2n+1). This same isomorphism emerged in recent work of Mikhaylov and Witten in the context of string theory as a T-duality composed with an S-duality. We construct similar Hopf superalgebra isomorphisms for families of pairs of quantum affine superalgebras. An immediate consequence is that the representation categories of the quantum affine superalgebras in each pair are equivalent as strict tensor categories.
We introduce a new family of superalgebras which should be considered as a super version of the Khovanov-Lauda-Rouquier algebras. Let $I$ be the set of vertices of a Dynkin diagram with parity. To this data, we associate a family of graded superalgebras, the quiver Hecke superalgebras. When there are no odd vertices, these algebras are nothing but the usual Khovanov-Lauda-Rouquier algebras. We then define another family of graded superalgebras, the quiver Hecke-Clifford superalgebras, and show that they are weakly Morita superequivalent to the quiver Hecke superalgebras. Moreover, we prove tha…
Associated to quantum affine general linear Lie superalgebras are two families of short exact sequences of representations whose first and third terms are irreducible: the Baxter TQ relations involving infinite-dimensional representations; the extended T-systems of Kirillov--Reshetikhin modules. We make use of these representations over the full quantum affine superalgebra to define Baxter operators as transfer matrices for the quantum integrable model and to deduce Bethe Ansatz Equations, under genericity conditions.
Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra
本文通过构造超 Yangian 模的 Gelfand-Tsetlin 基并给出 Drinfeld 生成元的显式作用，重新证明其不可约性，同时刻画了有限维不可约驯顺 Y(gl(1|1))-模恰为薄模，并将相应结论推广到 A 型量子仿射超代数。
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.
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.
We study representations of the quantum affine superalgebra associated with a general linear Lie superalgebra. In the spirit of Hernandez-Jimbo, we construct inductive systems of Kirillov-Reshetikhin modules based on a cyclicity result that we established previously on tensor products of these modules, and realize their inductive limits as modules over its Borel subalgebra, the so-called $q$-Yangian. A new generic asymptotic limit of the same inductive systems is proposed, resulting in modules over the full quantum affine superalgebra. We derive generalized Baxter's relations in the sense of F…
On the theory of $q$-characters for quantum affine superalgebras of type $A$
本文提出了一种用于量子仿射超代数类型A有限维单模的q-特征算法，并建立了与变形Cartan矩阵相关的q-特征理论。
We describe the category of integrable sl(1|n)^ -modules with the positive central charge and show that the irreducible modules provide the full set of irreducible representations for the corresponding simple vertex algebra.
Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$
本文引入量子仿射超代数的量子 Berezinian，证明其系数属于中心，构造另一族中心元素，并证明多个矩阵定理的类似物。
The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra $\U_q(\hat{\mathfrak{sl}}(M|N))$ at generic $q$. Any such module is shown to be a highest weight or lowest weight module with respect to one of the two natural triangular decompositions of the quantum affine superalgebra depending on whether the level of the module is zero or not. Furthermore, integrable $\U_q(\hat{\mathfrak{sl}}(M|N))$-modules at nonzero levels exist only if $M$ or $N$ is $1$.
In this paper we use the Etingof-Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld-Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight mod…
We construct the finite dimensional simple integral modules for the (degenerate) affine Hecke-Clifford algebra (AHCA). Our construction includes an analogue of Zelevinsky's segment representations, a complete combinatorial description of the simple calibrated modules, and a classification of the simple integral modules. Additionally, we construct an analogue of the Arakawa-Suzuki functor for the Lie superalgebra of type Q.
We describe the structure of a Verma module with a generic highest weight at the critical level over a symmetrizable affine Lie superalgebra not of the type A(2k,2l)^{(4)}. We obtain the character formula for a simple module with a generic highest weight at the critical level conjectured by V. G. Kac and D. A. Kazhdan.
We prove an explicit condition on the level $k$ for the irreducibility of a vacuum module $V^{k}$ over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level $k$ is a non-negative integer.
A Vogan diagram is a Dynkin diagram with a Cartan involution of twisted affine superlagebras based on maximally compact Cartan subalgebras. This article construct the Vogan diagrams of twisted affine superalgebras. It is a part of completion of classification of vogan diagrams to superalgebras cases.
We use the RTT realization of the quantum affine superalgebra associated with the Lie superalgebra $\mathfrak{gl}(M,N)$ to study its finite-dimensional representations and their tensor products. In the case $\mathfrak{gl}(1,1)$, the cyclicity condition of tensor products of finite-dimensional simple modules is determined completely in terms of zeros and poles of rational functions. This in turn induces cyclicity of some particular tensor products of Kirillov-Reshetikhin modules related to $\mathfrak{gl}(M,N)$.
The study of denominator identities for Lie superalgebras was recently developed by M. Gorelik, V.G. Kac, P.Moseneder Frajria, I. Musson, P. Papi, M. Wakimoto and the author. In this paper we generalize these identities to the twisted affine case, completing the result to all affine Lie superalgebras.
Oscillator representations of quantum affine orthosymplectic superalgebras
本文引入D型量子仿射超代数的q-振子表示范畴并构造新族不可约表示，基于超对偶理论证明其插值两类不可约表示，建立了Howe对偶对应的量子仿射类比。
We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type "modifier" is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak{g}$ is non-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $…
On results of certain modules over untwisted affine Liesuperalgebras
为完成仿射李超代数上模的刻画，本文收集了扭仿射李超代数表示中对非扭仿射李超代数成立的已知结果，其证明仅需微小修改，供后续使用。
In this paper, we used the free fields of Wakimoto to construct a class of irreducible representations for the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. The structures of the representations over the general linear Lie superalgebra and the special linear Lie superalgebra are studied in this paper. Then we extend the construction to the affine Kac-Moody Lie superalgebra $\widehat{\mathfrak{gl}_{m|n}}(\mathbb{C})$ on the tensor product of a polynomial algebra and an exterior algebra with infinitely many variables involving one parameter $μ$, and we also obtain the necess…
Affine super Yangians and deformed double current superalgebras
该研究将Schur-Weyl对偶相关结果推广到特殊线性李超代数$\mathfrak{sl}_{m|n}$的超Yangian及其仿射版本，还首次提出该代数的形变双流超代数并完成同类推广。
For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.
Irreducible nonzero level modules with finite-dimensional weight spaces are studied for non-twisted affine Lie superalgebras. A complete classification is obtained for superalgebras A(m,n)^ and C(n)^. In other cases the classification problem is reduced to the classification of cuspidal modules over finite-dimensional cuspidal Lie superalgebras described by Dimitrov, Mathieu and Penkov. It is also shown that any irreducible weakly integrable (in the sense of Kac and Wakimoto) module is highest weight.
Harish-Chandra modules for map and affine Lie superalgebras
本文分类了基本经典映射超代数上具有有限权重的单模，证明其由尖点有界模抛物诱导且同构于赋值模，并将结果推广到仿射 Kac-Moody 李超代数，构造零级有界单模并给出分类猜想。
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。
We prove the Weyl denominator identity for the affine Lie superalgebra gl(2|2)^ conjectured by V. Kac and M. Wakimoto. As it was pointed out in their paper, this gives another proof of Jacobi identity for the number of presentations of a given integer as a sum of eight squares.
Weyl denominator identity for the affinization of a basic Lie superalgebra with a non-zero Killing form was formulated by Kac and Wakimoto and was proven by them in defect one case. In this paper we prove this identity.
$(d,σ)$-twisted Affine-Virasoro superalgebras
本文引入了带有偶导子和有限阶自同构的 $(d,σ)$-扭曲 Affine-Virasoro 超代数及其泛中心扩张，并利用权函子和 $A$-覆盖方法完成了对其单尖点模与单拟有限模的分类。
Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms
本文在特征不为2的域上，基于可分闭包上的表示及其Galois扭转，分类了仿射群超概形的不可约表示，计算了其自同态的除超代数，并利用Shibata的Borel--Weil理论给出了特定条件下拟约化代数超群的数值结论。
Affine Super Yangians and Rectangular $W$-superalgebras
该研究构造了仿射超 Yangian 到矩形 $W$-超代数泛包络代数的同态，并给出其像稠密的参数条件。
Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are constructed explicitly in terms of free bosonic fields.
A note on Appell's functions related to the denominators of affine Lie superalgebras $\widehat{sl}(2|1)$ and $\widehat{osp}(3|2)$
本文研究最简 Appell 函数，给出其显式公式、模变换性质和渐近行为，与仿射李超代数分母相关。
In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares …
We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{s\ell}_{2|1}$ (resp. $\hat{ps\ell}_{2|2}$) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-\frac{2m+2}{M})$, where $m \in \mathbb{Z}_{\geq 0}$, $M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m+2,M)=1$ if $m>0$ (resp. $6(\frac{m}{M}-1)$, where $m \in \m…
Ian Grojnowski has developed a purely algebraic way to connect the representation theory of affine Hecke algebras at an (l+1)-th root of unity to the highest weight theory of the affine Kac-Moody algebra of type A_l^(1). The present article is devoted to extending Grojnowski's machinery to the twisted case: we replace the affine Hecke algebras with the affine Hecke-Clifford superalgebras of Jones and Nazarov, and the Kac-Moody algebra A_l^(1) with the twisted algebra A_2l^(2). In particular, we obtain an algebraic construction purely in terms of the representation theory of Hecke-Clifford supe…
We propose a generalization of Schramm-Loewner evolution (SLE) that has internal degrees of freedom described by an affine Lie superalgebra. We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra whose underlying finite dimensional Lie superalgebra is basic classical type, and write down stochastic differential equations on internal degrees of freedom in case that the corresponding affine Lie superalgebra is $\widehat{\mathfrak{osp}(1|2)}$. We also demonstrate computation of local martingales associated with the solution from a representation …
The purpose of this article is to investigate relations between W-superalgebras and integrable super-Hamiltonian systems. To this end, we introduce the generalized Drinfel'd-Sokolov (D-S) reduction associated to a Lie superalgebra $g$ and its even nilpotent element $f$, and we find a new definition of the classical affine W-superalgebra $W(g,f,k)$ via the D-S reduction. This new construction allows us to find free generators of $W(g,f,k)$, as a differential superalgebra, and two independent Lie brackets on $W(g,f,k)/\partial W(g,f,k).$ Moreover, we describe super-Hamiltonian systems with the P…
The universal $R$-matrix of the quantum affine superalgebra associated to the Lie superalgebra $\mathfrak{gl}(1,1)$ is realized as the Casimir element of certain Hopf pairing, based on the explicit coproduct formula of all the Drinfeld loop generators.
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。
In this paper, we classify all simple Harish-Chandra modules over the super affine-Virasoro algebra $\widehat{\mathcal{L}}=\mathcal{W}\ltimes(\mathfrak{g}\otimes \mathcal{A})\oplus \mathbb{C}C$, where $\mathcal{A}=\mathbb{C}[t^{\pm 1}]\otimesΛ(1)$ is the tensor superalgebra of the Laurent polynomial algebra in even variable $t$ and the Grassmann algebra in odd variable $ξ$, $\mathcal{W}$ is the Lie superalgebra of superderivations of $\mathcal{A}$, and $\mathfrak{g}$ is a finite-dimensional perfect Lie superalgebra.
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