D. Arnaudon, N. Crampe', L. Frappat, E. Ragoucy
We present the Drinfel'd realisation of the super Yangian Y(osp(1|2)), including the explicit expression for the coproduct. We show in particular that it is necessary to introduce supplementary Serre relations. The construction of its quantum double is carried out. This allows us to give the universal R-matrix of DY(osp(1|2)).
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Gaussian generators for the Yangian associated with the Lie superalgebra $\mathfrak{osp}(1|2m)$
本文利用 R-矩阵表示中的高斯分解，结合超-Yangian 嵌入定理，给出了正交辛李超代数 osp(1|2m) 的 Yangian 的一种新表示及其高斯生成元定义关系。
Representations of the Yangians associated with Lie superalgebras ${\frak{osp}}(1|2n)$
本文借助 Drinfeld 多项式并基于 $n=1$ 的已有结果，完成了与李超代数 ${\ rak{osp}}_{1|2n}$ 相关 Yangian 的有限维不可约表示分类。
We give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n) and osp(m|2n).
Representations of the super-Yangian of type $D(n,m)$
本文研究D(n,m)型超Yangian的有限维不可约表示分类，给出必要条件并猜想其充分性，对线性最高权表示证明猜想，并构造osp(2|2)与gl(1|2)对应Yangian的同构。
A Drinfeld-type presentation of the orthosymplectic Yangians
本文通过对正辛李超代数 Yangian 的 $R$-矩阵表示作 Gauss 分解，并借助超版嵌入定理，构造并证明了其 Drinfeld 型表示。
We show that some finite W-superalgebras based on gl(M|N) are truncation of the super-Yangian Y(gl(M|N)). In the same way, we prove that finite W-superalgebras based on osp(M|2n) are truncation of the twisted super-Yangians Y(gl(M|2n))^{+}. Using this homomorphism, we present these W-superalgebras in an R-matrix formalism, and we classify their finite-dimensional irreducible representations.
We formulate the Bethe Ansatz equations for the open super spin chain based on the super Yangian of osp(M|2n) and with diagonal boundary conditions. We then study the bulk and boundary scattering of the osp(1|2n) open spin chain.
We present a classification of diagonal, antidiagonal and mixed reflection matrices related to Yangian and super-Yangian R matrices associated to the infinite series so(m), sp(n) and osp(m|n). We formulate the analytical Bethe Ansatz resolution for the so(m) and sp(n) open spin chains with boundary conditions described by the diagonal solutions.
We construct universal Drinfel'd twists defining deformations of Hopf algebra structures based upon simple Lie algebras and contragredient simple Lie superalgebras. In particular, we obtain deformed and dynamical double Yangians. Some explicit realisations as evaluation representations are given for sl(N), sl(1|2) and osp(1|2).
Representations of the super-Yangian of type $B(n,m)$
本文研究$B(n,m)$型超Yangian的有限维不可约表示，通过嵌入定理和Yangian型奇反射推导必要条件，提出充分性猜想并在$n=1$、任意$m\geqslant1$时证明。
Representations of the super Yangians of types $A$ and $C$
本文分类了与正交辛李超代数 osp(2|2n) 相关联的超Yangian的有限维不可约表示，并基于Schur–Sergeev对偶给出了 gl(m|n) Yangian多项式求值模的R-矩阵构造。
Triangular deformation of the orthosymplectic Lie superalgebra osp(1|4) is defined by chains of twists. Corresponding classical r-matrix is obtained by a contraction procedure from the trigonometric r-matrix. The carrier space of the constant r-matrix is the Borel subalgebra.
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。
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。
Using the corepresentation of the quantum supergroup OSp_q(1/2) a general method for constructing noncommutative spaces covariant under its coaction is developed. In particular, a one-parameter family of covariant algebras, which may be interpreted as noncommutative superspheres, is constructed. It is observed that embedding of the supersphere in the OSp_q(1/2) algebra is possible. This realization admits infinitesimal characterization a la Koornwinder. A covariant oscillator realization of the supersphere is also presented.
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)^…
Drinfeld super Yangian of the exceptional Lie superalgebra $D(2,1;λ)$
本文为缺乏经典代数对应物的例外李超代数 $D(2,1;λ)$ 建立了首个 Drinfeld 超杨代数严格框架，显式给出其 Drinfeld 展示、PBW 基构造及 Hopf 超代数结构。
We consider a version of the non-linear Schrödinger equation with M bosons and N fermions. We first solve the classical and quantum versions of this equation, using a super-Zamolodchikov-Faddeev (ZF) algebra. Then we prove that the hierarchy associated to this model admits a super-Yangian Y(gl(M|N)) symmetry. We exhibit the corresponding (classical and quantum) Lax pairs. Finally, we construct explicitly the super-Yangian generators, in terms of the canonical fields on the one hand, and in terms of the ZF algebra generators on the other hand. The latter construction uses the well-bred operator…
Twisted super Yangians of type AIII and their representations
本文研究 Molev-Ragoucy 反射代数的超类似物（AIII 型 twisted super Yangians），在特定条件下分类其有限维不可约表示，并建立其与 BC 型退化仿射 Hecke 代数的 Schur-Weyl 型对偶。
Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{m|n}$
本文研究李超代数 $\mathfrak{gl}_{m|n}$ 的双 Yangian，证明其 Poincaré-Birkhoff-Witt 定理、构造量子缩并系数形式的中心元，并据此给出 level 0 反射代数的生成元—关系表述。
Orthosymplectic Yangians
研究正交辛超杨氏代数，给出任意奇偶序列下的Drinfeld实现，推广了非超BCD型、标准奇偶序列及超A型的结果。
Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
本研究推广已有方法，通过将扭与非扭量子仿射超代数及对应Yangian的融合顶点模型QQ关系和T函数约化为$U_{q}(gl(M|N)^{(1)})$的对应形式，重现并推导了多种表示下T函数的表达式，为超自旋链贝特ansatz提供统一路径。
The Racah problem for the quantum superalgebra $\mathfrak{osp}_{q}(1|2)$ is considered. The intermediate Casimir operators are shown to realize a $q$-deformation of the Bannai-Ito algebra. The Racah coefficients of $\mathfrak{osp}_q(1|2)$ are calculated explicitly in terms of basic orthogonal polynomials that $q$-generalize the Bannai-Ito polynomials. The relation between these $q$-deformed Bannai-Ito polynomials and the $q$-Racah/Askey-Wilson polynomials is discussed.
We construct a minimalistic presentation of Drinfeld super Yangians in the case of special linear superalgebra associated with an arbitrary Dynkin diagram. This gives us a possibility to introduce Hopf superalgebra structure on Drinfeld super Yangians. Using complete Weyl group we classify Drinfeld super Yangians endowed with mentioned Hopf superalgebra structures. Also it is constructed an isomorphism between completions of Drinfeld super Yangians and quantum loop superalgebras.
We construct highest-weight modules and a Yangian extension of the centrally extended sl(1|1)^2 superalgebra, that is a symmetry of the worldsheet scattering associated with the AdS3/CFT3 duality. We demonstrate that the R-matrix intertwining atypical modules is of a trigonometric type. We also consider a quantum deformation of this superalgebra, its modules, and obtain a quantum affine extension of the Drinfeld-Jimbo type that describes a deformed worldsheet scattering.
Following V. Toledano-Laredo and S. Gautam approach we construct isomorphism between super $\hbar$-Yangian $Y_{\hbar}(A(m,n))$ of special linear superalgebra and quantum loop superalgebra $U_{\hbar}(LA(m,n))$.
The twisting function describing a nonstandard (super-Jordanian) quantum deformation of $osp(1|2)$ is given in explicite closed form. The quantum coproducts and universal R-matrix are presented. The non-uniqueness of the twisting function as well as two real forms of the deformed $osp(1|2)$ superalgebras are considered. One real quantum $osp(1|2)$ superalgebra is interpreted as describing the $κ$-deformation of D=1, N=1 superconformal algebra, which can be applied as a symmetry algebra of N=1 superconformal mechanics.
Yangian Double $DY(A(m,n))$ of Lie Superalgebra $A(m,n)$ is described in terms of generators and defining relations. It is proved triangular decomposition for Yangian $Y(A(m,n))$ and its quantum double $DY(A(m,n))$ as a corollary of PBW theorem. It is introduced the normally ordered bases in Yangian and its dual Hopf superalgebra in quantum double. It is calculated the pairing formulas between the elements of its bases. It is received the formula of Universal $R$-matrix of Yangian Double. The formula for Universal $R$-matrix of Yangian, which was introduced by V. Drinfel'd, is received also.
We give a new proof of the fact that the super Yangian of general linear Lie superalgebra is isomorphic to the finite W-superalgebra of the general linear Lie superalgebra associated to a rectangular nilpotent element.
Hopf superalgebra structure for Drinfeld super Yangian of Lie superalgebra $B(m,n)$
本文通过引入 Drinfeld 超 Yangian 的简化定义，为李超代数 $B(m,n)$ 构造了相对于所有可能 Borel 子代数选择的 Hopf 超代数结构。
Quantum Berezinian for the Twisted Super Yangian
本文针对扭超 Yangian 给出量子 Berezinian 的显式公式，构造其中心与特殊扭超 Yangian，并证明生成矩阵及量子 Berezinian 的量子 Sylvester 定理。
We extend Yangian double to super (or graded) case and give its Drinfel'd generators realization by Gauss decomposition.
We use the super version of RS construction method and Gauss decomposition to obtain Drinfeld's currents realization of super Yangian double $DY(gl(1|1))$ with center extension
For any fixed composition $μ$ of $M+N$ and any fixed $0^M1^N$-sequence $\mathfrak{s}$, we obtain a new presentation of the super Yangian $Y_{M|N}$ associated to the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$.
We introduce super Yangian double $DY_\hbar[gl(m|n)]$ and its central extension $\widehat{DY_\hbar[gl(m|n)]}$. We give their defining relations in terms of current generators and obtain Drinfeld comultiplication.
Super Yangians in characteristic $2$
本文在特征2的域上定义了超Yangian Y_{m|n}，并展示了其作为当前李代数gl_{m+n}[t]超通用 enveloping 代数的变形，同时描述了其中心结构。
The Yangian of the strange Lie superalgebras in Drinfel'd realization is defined. The current system generators and defining relations are described.
Associated to a composition of M and a composition of N, a new presentation of the super Yangian of the general linear Lie superalgebra $Y(gl_{M|N})$ is obtained.
The co-Lie structures compatible with the osp(2|2) Lie super algebra structure are investigated and found to be all of coboundary type. The corresponding classical r-matrices are classified into several disjoint families. The osp(1|2)+u(1) Lie super-bialgebras are also classified.
Parabolic presentations of the modular Super Yangian Y(M|N)
本文将超杨振子Y(M|N)的抛物型表示推广到正特征域，依赖于任意组合和01序列。
Let $e$ be an arbitrary even nilpotent element in the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$ and let $\mathcal{W}_e$ be the associated finite $W$-superalgebra. Let $Y_{m|n}$ be the super Yangian associated to the Lie superalgebra $\mathfrak{gl}_{m|n}$. A subalgebra of $Y_{m|n}$, called the shifted super Yangian and denoted by $Y_{m|n}(σ)$, is defined and studied. Moreover, an explicit isomorphism between $\mathcal{W}_e$ and a quotient of $Y_{m|n}(σ)$ is established.
We study the finite W-superalgebra $W_e$ associated to a nilpotent element $e$ in a general linear Lie superalgebra. Under certain restriction on the Jordan type of $e$, we give a realization of $W_e$ in terms of a quotient of a shifted super Yangian.
A triangular quantum deformation of $ osp(2/1) $ from the classical $r$-matrix including an odd generator is presented with its full Hopf algebra structure. The deformation maps, twisting element and tensor operators are considered for the deformed $ osp(2/1)$. It is also shown that its subalgebra generated by the Borel subalgebra is self-dual.
Based on Drinfeld realization of super Yangian Double DY(gl(1|1)), its pairing relations and universal R-matrix are given. By taking evaluation representation of universal R-matrix, another realization $L^{\pm}(u)$ of DY(gl(1|1)) is obtained. These two realizations of DY(gl(1|1)) are related by the supersymmetric extension of Ding-Frenkel map.
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.
Coproducts for affine super-Yangian and Weyl groupoid action
本研究定义了仿射特殊线性超代数对应的仿射超Yangian，证明其两种表示同构并诱导余乘法，构造Weyl广群的作用，得出不同单根系定义的超Yangian作为Hopf超代数同构的结论。
We study the center of the two-parameter quantum supergroup $U_{r,s}(osp(1,2))$ using the Dickson polynomial. We show that the Scasimir operator is completely determined by the $q$-deformed Chebychev polynomial, generalizing an earlier work of Arnaudon and Bauer.
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.
A note on the center of the super Yangian $Y_{M|N}(\mathfrak{s})$
本文针对任意固定 $0^M1^N$-序列对应的超 Yangian，利用抛物生成元构造了量子 Berezinian 的新公式，推广了基于 RTT 生成元和 Drinfeld 生成元的经典表达式。
A note on the center of the queer super Yangian ${\rm Y}(\mathfrak{q}_1)$
本文研究 queer 超 Yangian ${\rm Y}(\frak{q}_1)$ 的两类中心元，明确其生成级数间的显式关系，为该领域相关研究提供了具体结论。
From quantum loop superalgebras to super Yangians
本文将Drinfeld关于Yangian与量子loop代数关系的结论推广到超代数情形，建立了一般线性李超代数的量子loop超代数与超Yangian的联系，推导了对应量子loop超代数的PBW定理，并将论证拓展至扭超Yangian，引入了扭量子loop超代数作为单侧余理想。
We obtain a closed form expression of the universal T-matrix encapsulating the duality of the quantum superalgebra U_q[osp(1/2)] and the corresponding supergroup OSp_q(1/2). The classical q-->1 limit of this universal T-matrix yields the group element of the undeformed OSp(1/2) supergroup. The finite dimensional representations of the quantum supergroup OSp_q(1/2) are readily constructed employing the said universal T-matrix and the known finite dimensional representations of the dually related deformed U_q[osp(1/2)] superalgebra. Proceeding further, we derive the product law, the recurrence r…
Methods are developed for systematically constructing the finite dimensional irreducible representations of the super Yangian Y(gl(M|N)) associated with the Lie superalgebra gl(M|N). It is also shown that every finite dimensional irreducible representation of Y(gl(M|N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite dimensional are given.
We develop a generic reprersentation-independent contraction procedure for obtaining, for instance, $R_{\sf h}$ and $L$ operators of arbitrary dimensions for the quantized ${\cal U}_{\sf h}(osp(2|1))$ algebra corresponding to the classical $r_2$ matrix from the pertinent quantities of the standard q-deformed ${\cal U}_q(osp(2|1))$ algebra. Also the quantized ${\bf U_h}(osp(2|1))$ algebra corresponding to the classical $r_1$ matrix comprising of the generators of the classical $sl(2)$ algebra is obtained in terms of a nonlinear basis set.
Cartan-Weyl basis for the quantum affine superalgebra U_q(^osp(1|2)) is constructed in an explicit form.
Extending a recently proposed procedure of construction of various elements of diffential geometry on noncommutative algebras, we obtain these structures on noncommutative superalgebras. As an example, a quantum superspace covariant under the action of super-Jordanian $OSp_h(2/1)$ is studied. It is shown that there exist a two paramater family of torsionless connections, and the curvature computed from this family of connections is bilinear. It is also shown that the connections are not compatible with the metric.
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 discuss the integrability structure of deformed, four-dimensional N=4 super Yang-Mills theories using Yangians. We employ a recent procedure by Beisert and Roiban that generalizes the beta deformation of Lunin and Maldacena to produce N=1 superconformal gauge theories, which have the superalgebra SU(2,2|1)xU(1)xU(1). The deformed theories, including those with the more general twist, were shown to have retained their integrable structure. Here we examine the Yangian algebra of these deformed theories. In a five field subsector, we compute the two cases of SU(2)xU(1)xU(1)xU(1) and SU(2|1)xU(…
Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so contains a \textit{universal $R$-matrix} in the tensor product algebra which satisfies the Yang-Baxter equation. Applying the vector representation $π$, which acts on the vector module $V$, to one side of a universal $R$-matrix gives a Lax operator. In this paper a Lax operator is constructed for the $C$-type quantum superalgebras $U_q[osp(2|n)]$. This can in turn be used to find a solution to the Yang-Baxter equation acting on $V \otimes V \otimes W$ where $W$ is an arbitrary $U_q[osp(2|n)]$ module. The case $W=V$ is includ…
A note on the quantum Berezinian for the double Yangian of the Lie superalgebra $\mathfrak{gl}_{m|n}$
该研究将量子Berezinian概念推广到李超代数$\frak{gl}_{m|n}$的双Yangian ${\rm DY}(\frak{gl}_{m|n})$，证明其系数是该双Yangian中心的代数独立拓扑生成元。
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