Maria Gorelik, Vera Serganova
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.
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We study Duflo-Serganova functor for non-twisted affine Lie superalgebras and affine vertex superalgebras.
We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie 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.
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.
We prove a denominator identity for non-twisted affine Lie superalgebras with zero dual Coxeter number.
We classify integrable bounded simple weight modules over classical Lie superalgebras at infinity. We also study the categories of such modules, and we prove that for most of the classical Lie superalgebras at infinity the respective category is semisimple.
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.
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.
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。
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.
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 $1$.
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…
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 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.
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 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…
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。
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。
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.
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。
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…
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.
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.
We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}_{2|1}$ can be modified, using Zwegers' real analytic corrections, to form an $SL_2(\mathbf{Z})$-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for $\widehat{osp}_{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-\frac{2m+2}{M})$, where $m \…
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.
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.
Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategor…
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。
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.
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 introduce super Yangians of $\mathfrak{gl}(V),\mathfrak{sl}(V)$ (in the new Drinfeld realization) associated to all Dynkin diagrams of $\mathfrak{gl}(V)$, where $V$ is a finite-dimensional super vector space. We show that all of them are isomorphic to the super Yangians introduced by Maxim Nazarov, by identifying them with the corresponding RTT super Yangians. However, their "positive halves" are not pairwise isomorphic, and we obtain the shuffle algebra realizations of those. We adapt the latter to the trigonometric setup by obtaining the shuffle algebra realizations of the "positive halve…
In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules $L$ over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras $\mathfrak{g}$. The problems consists of two parts. First, it is the reduction of the problem to the $\overline{\mathfrak{g}}$-module $F(L)$, where $\overline{\mathfrak{g}}$ is the associated to $L$ integral Lie superalgebra and $F(L)$ is an integrable irreducible highest weight $\overline{\mathfrak{g}}$-module. Second, it is the computation of characters of integrable highest weight m…
For all almost affine (hyperbolic) Lie superalgebras, the defining relations are computed in terms of their Chevalley generators.
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 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.
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…
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.
Superalgebra deformations of web categories: Affine and cyclotomic webs
本文研究了与局部单位Frobenius超代数相关的仿射和循环网范畴的超代数变形，提出了装饰双余类图基并建立了相关基结果。
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.
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。
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.
Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in terms of hook Schur functions. We use the reduction procedure of $\hgltwo$ to $\hat{gl}_{n|n}$ to derive a character formula for a certain class of level 1 highest weight irreducible representations of $\hat{gl}_{n|n}$, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra $gl_{n|n}$. These modules turn out to form the complete s…
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…
Quasi-integrable modules over affine Lie superalgebras (Critical level)
本文研究了仿紧李超代数在临界水平下的模块表征，通过分析特定子代数的结构，完整描述了零水平下的准可积模块特性。
In this work infinitesimal deformations of the model filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebra have been studied. All the filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras can be obtained by means of infinitesimal deformations, hence the importance of these. Thus, in particular, we give a family of filiform $\mathbb{Z}_2 \times \mathbb{Z}_2$-color Lie superalgebras via linearly integrable deformations.
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 $\mathfrak{gl}(M,N)$.
We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W-algebras. We also describe the centralizer of a nilpotent even element and of an sl(2)-triple in gl(m|n) and osp(m|2n).
We obtain a complete classification of all finite-dimensional irreducible modules over classical map superalgebras, provide formulas for their (super)characters and a description of their extension groups. Furthermore, we describe the block decomposition of the category of finite-dimensional modules for such map superalgebras. As an application, we specialize our results to the case of loop superalgebras in order to obtain a classification of finite-dimensional irreducible modules and block decomposition of the category of finite-dimensional modules over affine Lie superalgebras.
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。
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。
Affine oriented Frobenius Brauer categories
本文为任意 Frobenius 超代数 A 构造有向及仿射有向 Frobenius Brauer 范畴，并定义其在 gl_{m|n}(A) 超模范畴上的自然作用，统一推广一般线性与 queer 李超代数情形。
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。
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