Yao-Zhong Zhang
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.
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We introduce the elliptic superalgebra $U_{q,p}(\hat{sl}(M|N))$ as one parameter deformation of the quantum superalgebra $U_q(\hat{sl}(M|N))$. For an arbitrary level $k \neq 1$ we give the bosonization of the elliptic superalgebra $U_{q,p}(\hat{sl}(1|2))$ and the screening currents that commute with $U_{q,p}(\hat{sl}(1|2))$ modulo total difference.
We give bosonizations of the superalgebras $U_q(\hat{sl}(N|1))$ and $U_{q,p}(\hat{sl}(N|1))$ for an arbitrary level $k \in {\bf C}$. We introduce the submodule by the $ξ$-$η$ system, that we call the $q$-Wakimoto realization.
We construct a free field realization of the quantum affine superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k \in {\mathbb C}$.
We construct the screening currents of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k \neq -N+1$. We show that these screening currents commute with the superalgebra modulo total difference. We propose bosonizations of the vertex operators by using the screening currents. We check that these vertex operators are the intertwiners among the Fock-Wakimoto representation and the typical representation for rank $N \leq 4$.
We introduce the quasi-Hopf superalgebras which are $Z_2$ graded versions of Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting the normal quantum supergroups by twistors which satisfy the graded shifted cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the supersymmetric case. Two types of elliptic quantum supergroups are defined, that is the face type $B_{q,λ}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-…
We obtain Drinfel'd's realization of quantum affine superalgebra $U_q\hat{(gl(1|1))}$ based on the super version of RS construction method and Gauss decomposition.
A bosonization of the quantum affine superalgebra $U_q(\widehat{sl}(M|N))$ is presented for an arbitrary level $k \in {\bf C}$.The Wakimoto realization is given by using $ξ-η$ system. The screening operators that commute with $U_q(\widehat{sl}(M|N))$ are presented for the level $k \neq -M+N$. New bosonization of the affine superalgebra $\widehat{sl}(M|N)$ is obtained in the limit $q \to 1$.
In this paper, we prove classical affine W-algebras associated to Lie superalgebras (W-superalgebras) can be constructed in two different ways: via affine classical Hamiltonian reductions and via taking quasi-classical limits of quantum affine W-superalgebras. Also, we show that a classical finite W-superalgebra can be obtained by a Zhu algebra of a classical affine W-superalgebra. Using the definition by Hamiltonian reductions, we find free generators of a classical W-superalgebra associated to a minimal nilpotent. Moreover, we compute generators of the classical W-algebra associated to $spo(…
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.
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.
In this paper, we present the hidden symmetry behind the Faddeev-Reshetikhin-Takhtajan-Semenov-Tian-Shansky realization of quantum affine superalgebras $U_q(\hat {osp}(1,2))$ and add the q-Serre relation to the Drinfeld realization of $U_q(\hat {osp}(1,2))$ (math.QA/9712011) derived from the FRTS realization.
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.
A level-one representation of the quantum affine superalgebra $\U_q(\hat{\frak{sl}}(M+1|N+1))$ and vertex operators associated with the fundamental representations are constructed in terms of free bosonic fields. Character formulas of level-one irreducible highest weight modules of $\U_q(\hat{\frak{sl}}(2|1))$ are conjectured.
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…
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 introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.
In this paper, we extend the generalization of Drinfeld realization of quantum affine algebras to quantum affine superalgebras with its Drinfeld comultiplication and its Hopf algebra structure, which depends on a function $g(z)$ satisfying the relation: $g(z)=g(z^{-1})^{-1}.$ In particular, we present the Drinfeld realization of $U_q(\hat {\frak osp}(1,2))$ and its Serre relations.
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 study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic $\hat{\cal R}$-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic $\hat{\cal R}$-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such $\hat{\cal R}$ operators in explicit form for special cases of the $osp(n|2m)$ algebra, in particular for a few low-rank cases. We also propose a novel, simpl…
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.
We give shuffle algebra realization of positive part of quantum affine superalgebra $U_{v}(\widehat{\mathfrak{D}}(2,1;θ))$ associated to any simple root systems. We also determine the shuffle algebra associated to $\widehat{\mathfrak{sl}}(2|1)$ with odd root system when $v$ is a primitive root of unity of even order, generalizing results in \cite{FJMMT03}.
In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld's realizations of $U_h({\hat {sl}}(m|n)^{(1)})$.
We construct Drinfeld realisations for the quantum affine superalgebras associated with the osp(1|2n)^{(1)}, Sl(1|2n)^{(2)} and osp(2|2n)^{(2)} series of affine Lie superalgebras.
The minimal irreducible representations of $U_q[gl(m|n)]$, i.e. those irreducible representations that are also irreducible under $U_q[osp(m|n)]$ are investigated and shown to be affinizable to give irreducible representations of the twisted quantum affine superalgebra $U_q[gl(m|n)^{(2)}]$. The $U_q[osp(m|n)]$ invariant R-matrices corresponding to the tensor product of any two minimal representations are constructed, thus extending our twisted tensor product graph method to the supersymmetric case. These give new solutions to the spectral-dependent graded Yang-Baxter equation arising from $U_q…
Affine super Yangians and non-rectangular $W$-superalgebras
本文通过构造 A 型仿射超 Yangian 的四个边收缩，建立了其到 A 型非矩形 W-超代数泛包络代数的同态。
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。
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be deformations of their classical counterparts. It is also shown that Jimbo type quantum affine superalgebras can be obtained by deforming universal enveloping algebras of ordinary (i.e., non-graded) affine algebras supplemented by certain parity operators.
Semisimplicity of module categories of certain affine vertex operator superalgebras
本文证明了某些仿射顶点算子超代数在共形但非容许水平上的 Kazhdan-Lusztig 范畴是半单的，并由此得到其有限长度广义模范畴具有辫子张量范畴结构。
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 …
Drinfeld realisations are constructed for the quantum affine superalgebras of the series ${\rm\mathfrak{osp}}(1|2n)^{(1)}$,${\rm\mathfrak{sl}}(1|2n)^{(2)}$ and ${\rm\mathfrak{osp}}(2|2n)^{(2)}$. By using the realisations, we develop vertex operator representations and classify the finite dimensional irreducible representations for these quantum affine superalgebras.
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.
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.
We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.
Cartan-Weyl basis for the quantum affine superalgebra U_q(^osp(1|2)) is constructed in an explicit form.
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 …
Using super RS construction method and Gauss decomposition, we obtain Drinfeld's currents realization of two-parameter quantum affine superalgebra $U_{p,q}\widehat{(gl(1|1))}$ and get co-product structure for these currents.
We describe the realization of the super Reshetikhin-Semenov-Tian-Shansky (RS) algebra in quantum affine superalgebras, thus generalizing the approach of Frenkel-Reshetikhin to the supersymmetric (and twisted) case. The algebraic homomorphism between the super RS algebra and the Drinfeld current realization of quantum affine superalgebras is established by using the Gauss decomposition technique of Ding-Frenkel. As an application, we obtain Drinfeld realization of quantum affine superalgebra $U_q[osp(1|2)^{(1)}]$ and its degeneration -- central extended super Yangian double $DY_\hbar[osp(1|2)^…
We give a realization of the quantum affine Lie superalgebras U_q(A(M,N))^(1) in terms of anyons defined on a one or two-dimensional lattice, the deformation parameter q being related to the statistical parameter $ν$ of the anyons by q = exp(iπν). The construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of the quantum superalgebras U_q(A(M,N)) is obtained.
The type-I simple Lie-superalgebras are $sl(m|n)$ and $osp(2|2n)$. We study the quantum deformations of their untwisted affine extensions $U_q(sl(m|n)^{(1)})$ and $U_q(osp(2|2n)^{(1)})$. We identify additional relations between the simple generators (``extra $q$-Serre relations") which need to be imposed to properly define $\uqgh$ and $U_q(osp(2|2n)^{(1)})$. We present a general technique for deriving the spectral parameter dependent R-matrices from quantum affine superalgebras. We determine the R-matrices for the type-I affine superalgebra $U_q(sl(m|n)^{(1)})$ in various representations, ther…
We construct a bosonization of the quantum superalgebra $U_q(\hat{sl}(N|1))$ for an arbitrary level $k$. We construct the screening that commutes with the quantum superalgebra for an arbitrary level $k \neq -N+1$. We propose a bosonization of the vertex operator that gives the intertwiner among the Wakimoto realization and the typical representation.
We compute an universal weight function (off-shell Bethe vectors) in any representation with a weight singular vector of the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in arXiv:math/0610398.
Modules over affine Lie superalgebras ${\cal G}$ are studied, in particular, for ${\cal G}=\hat{OSP(1,2)}$. It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most importance are the generalized Kac-Kazhdan formula and the Malikov-Feigin-Fuchs construction, which give the weights and the explicit form of the singular vectors in the Verma module over affine Kac-Moody superalgebras. We have also considered the decomposition of the admissible representation of $\hat{OSP(1,2)}$ into that of $\hat{SL(2)}\otimes$Virasoro alge…
This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra $V_k(\mathfrak g)$ where $\mathfrak g=\mathfrak g_{\bar 0}\oplus \mathfrak g_{\bar 1}$ is a basic classical simple Lie superalgebras. Let $\mathcal V_k (\mathfrak g_{\bar 0})$ be the subalgebra of $V_k(\mathfrak g)$ generated by $\mathfrak g_{\bar 0}$. We first classify all levels $k$ for which the embedding $\mathcal V_k (\mathfrak g_{\bar 0})$ in $V_k(\mathfrak g)$ is conformal. Next we prove that, for a…
By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain Drinfeld current realization for quantum affine superalgebra $U_q[gl(m|n)^{(1)}]$. We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.
Intertwiners of representations of exceptional type quantum affine superalgebras
研究例外型量子仿射超代数最小非平凡不可约表示，给出D$_{2\vert 1;\alpha}$、F$_{3\vert 1}$、G$_{2\vert 1}$型在两种实现下的显式公式，借此得到相应$R$ - 矩阵的显式表达式。
We derive from the super RS algebra the Drinfeld basis of the twisted quantum affine superalgebra $U_q[osp(2|2)^{(2)}]$ by means of the Gauss decomposition technique. We explicitly construct a nonclassical level-one representation of $U_q[osp(2|2)^{(2)}]$ in terms of two $q$-deformed free boson fields.
We continue investigation of the universal weight function for the quantum affine algebra $U_q(\hat{\mathfrak{gl}}_N)$ started in arXiv:math/0610517 and arXiv:0711.2819. We obtain two recurrence relations for the universal weight function applying the method of projections developed in arXiv:math/0610398. On the level of the evaluation representation of $U_q(\hat{\mathfrak{gl}}_N)$ we reproduce both recurrence relations for the off-shell Bethe vectors calculated in arXiv:math/0702277 using combinatorial methods.
We give a precise expression for the universal weight function of the quantum affine algebra $U_q(\hat{sl}_3)$. The calculations use the technique of projecting products of Drinfeld currents on the intersections of Borel subalgebras.
In this paper, a notion of affine walled Brauer-Clifford superalgebras $BC_{r, t}^{\rm aff} $ is introduced over an arbitrary integral domain $R$ containing $2^{-1}$. These superalgebras can be considered as affinization of walled Brauer superalgebras in \cite{JK}. By constructing infinite many homomorphisms from $BC_{r, t}^{\rm aff}$ to a class of level two walled Brauer-Clifford superagebras over $\mathbb C$, we prove that $BC_{r, t}^{\rm aff} $ is free over $R$ with infinite rank. We explain that any finite dimensional irreducible $BC_{r, t}^{\rm aff} $-module over an algebraically closed f…
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。
We determine the exchange relations of the level-one q-vertex operators of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$. We study in details the level-one irreducible highest weight representations of $U_q[\hat{gl(2|2)}]$, and compute the characters and supercharacters associated with these irreducible modules.
W-algebras as conformal extensions of affine VOAs
研究从仿射顶点代数到顶点超代数的顶点算子超代数同态，给出其为共形及满射的准则，应用于 W - 代数和 W - 超代数，得到多种坍缩或扩张的例子，特别是非允许层级单 W - 代数坍缩为允许层级仿射顶点代数的例子。
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。
Braid group action and quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$
本文针对类型 $\mathfrak{osp}(2m+1|2n)$ 的量子仿射超代数，通过引入辫群作用定义量子根向量，提出了一种高效验证 Drinfeld-Jimbo 表示与 Drinfeld 表示同构的方法。
$(d,σ)$-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。
Commuting Subalgebras of Affine Super Yangians Arising from Edge Contractions
本文研究了仿射超杨代数边收缩的交换性，并应用该结果建立了与W超代数相关的一系列同态关系。
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 investigate the Lie algebra structures of weight one subspaces of $C_2$-cofinite vertex operator superalgebras. We also show that for any positive integer $k$, vertex operator superalgebras $L_{sl(1|n+1)}(k,0)$ and $L_{osp(2|2n)}(k,0)$ have infinitely many irreducible admissible modules. As a consequence, we give a proof of the fact that $L_{\mathfrak g}(k,0)$ is $C_2$-cofinite if and only if $\mathfrak g$ is either a simple Lie algebra, or $\mathfrak g=osp(1|2n)$, and $k$ is a nonnegative integer. As an application, we show that $L_{G(3)}(1,0)$ is a vertex operator superalge…
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