D. Arnaudon, J. Avan, N. Crampe', A. Doikou, L. Frappat, E. Ragoucy
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.
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We use the Dunkl operator approach to construct one dimensional integrable models describing N particles with internal degrees of freedom. These models are described by a general Hamiltonian belonging to the center of the Yangian or the reflection algebra, which ensures that they admit the corresponding symmetry. In particular, the open problem of the symmetry is answered for the B_N-type Sutherland model with spin and a generalized B_N-type nonlinear Schrodinger Hamiltonian.
The odd reflections are an effective tool in the Lie superalgebra representation theory, as they relate non-conjugate Borel subalgebras. We introduce analogues of the odd reflections for the Yangian ${\rm Y}({\frak gl}_{m|n})$ and use them to produce a transition rule for the parameters of the highest weight modules corresponding to a change of the parity sequence. This leads to a description of the finite-dimensional irreducible representations of the Yangian associated with an arbitrary parity sequence.
It is known that integrable models associated to rational $R$ matrices give rise to certain non-abelian symmetries known as Yangians. Analogously `boundary' symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the $N$ site $gl(n)$ open quantum spin chain. In particular, the open spin chain with two distinct types of bo…
We present the `Heisenberg picture' of the reflection algebra by explicitly constructing the boundary Yangian symmetry of an AdS/CFT superstring which ends on a boundary with non-trivial degrees of freedom and which preserves the full bulk Lie symmetry algebra. We also consider the spectrum of bulk and boundary states and some automorphisms of the underlying algebras.
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。
With any involutive anti-algebra and coalgebra automorphism of a quasitriangular bialgebra we associate a reflection equation algebra. A Hopf algebraic treatment of the reflection equation of this type and its universal solution is given. Applications to the twisted Yangians are considered.
We study a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in Cartan's classification. These algebras can be regarded as coideal subalgebras of the extended Yangian for orthogonal or symplectic Lie algebras. They can also be presented as quotients of a reflection algebra by additional symmetry relations. We prove an analogue of the Poincare-Birkoff-Witt Theorem, determine their centres and study also extended reflection algebras.
For the classical principal chiral model with boundary, we give the subset of the Yangian charges which remains conserved under certain integrable boundary conditions, and extract them from the monodromy matrix. Quantized versions of these charges are used to deduce the structure of rational solutions of the reflection equation, analogous to the 'tensor product graph' for solutions of the Yang-Baxter equation. We give a variety of such solutions, including some for reflection from non-trivial boundary states, for the SU(N) case, and confirm these by constructing them by fusion from the basic s…
We present a generalization the G. Letzter's theory of quantum symmetric pairs of semisimple Lie algebras for the case of quantum affine algebras. We then study solutions of the reflection equation for the quantum affine algebras sl(2) and gl(1|1) and their Yangian limit for singlet (diagonal) and vector (non-diagonal) boundary conditions. We construct the corresponding quantum affine coideal subalgebras that are based on the quantum symmetric pairs, and the (generalized) twisted Yangians.
A note on odd reflections of super Yangian and Bethe ansatz
本文从新视角重新推导超 Yangian 的奇反射转换规则，建立其与 XXX 型 Bethe ansatz 费米子再生程序的联系，并给出 q-特征标变化算法及斜表示的显式计算。
In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra. In this paper we analyse its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, rece…
Double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$, II: Quantum currents
本文研究李超代数$\frak{gl}_{M|N}$的双重杨代数与反射代数，构造临界水平双重杨代数的中心元，推广Etingof-Kazhdan构造引入量子顶点超代数结构，证明反射代数限制模的准模性质。
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…
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。
We give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n) and osp(m|2n).
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.
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。
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。
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。
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。
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)).
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。
The notion of compatible braidings was introduced by Isaev, Ogievetsky and Pyatov. On the base of this notion they defined certain quantum matrix algebras generalizing the RTT algebras and Reflection Equation ones. They also defined analogs of some symmetric polynomials in these algebras and showed that these polynomials generate commutative subalgebras, called Bethe. By using a similar approach we introduce certain new algebras called generalized Yangians and define analogs of some symmetric polynomials in these algebras. We claim that they commute with each other and thus generate a commutat…
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。
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。
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.
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 $U_{\hbar}(LA(m,n))$.
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 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.
We extend Yangian double to super (or graded) case and give its Drinfel'd generators realization by Gauss decomposition.
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(…
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
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。
We consider boundary scattering for a semi-infinite one-dimensional deformed Hubbard chain with boundary conditions of the same type as for the Y=0 giant graviton in the AdS/CFT correspondence. We show that the recently constructed quantum affine algebra of the deformed Hubbard chain has a coideal subalgebra which is consistent with the reflection (boundary Yang-Baxter) equation. We derive the corresponding reflection matrix and furthermore show that the aforementioned algebra in the rational limit specializes to the (generalized) twisted Yangian of the Y=0 giant graviton.
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.
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。
Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$
本文为仿射超杨代数 $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$ 构造了 cut-and-join 算子及其本征函数超 Schur 函数，并推广了 Pieri 规则、hook 公式和 Cauchy 公式，同时探讨了其弦论起源。
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.
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。
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 $\mathfrak{gl}_{M|N}$.
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。
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.
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.
Jacobi-Trudi identity and Drinfeld functor for super Yangian
本文证明超 Yangian 对应的量子 Berezinian 可表示为两个差分算子之比，并补全了其表示论中的 q-特征标理论、Jacobi-Trudi 恒等式、Drinfeld 函子等关键工具。
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。
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。
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。
Towards the theory of Yangians
本文综述 Yangian 作为弦论启发模型中隐藏对称性的理论框架，介绍箭图分类、环面 Calabi-Yau 构造及类 ADHM 模空间中的瞬子跃迁。
Representations of shifted super Yangians and finite $W$-superalgebras of type A
本文研究了移位超 Yangians 和类型 A 的有限 W-超代数的表示理论，给出了不可约模有限维性的判据，并为有限 W-超代数的 Verma 模提供了显式 Gelfand-Tsetlin 特征公式。
Twisted super Yangians of quasi-split type A
本文研究了准分裂型A的扭曲超Yangian代数，通过Gauss分解证明其与特殊扭曲超Yangian同构，并推广了非超情况下的已知结果。
Algebraic structures in set-theoretic Yang-Baxter & reflection equations
该研究围绕集合论Yang-Baxter方程与反射方程的解展开，梳理brace代数性质，分析对合解的参数化解与量子代数、Drinfeld扭与Yangian的关联，推导反射相关的代数关系并证明特殊反射映射下存在反射有限子代数。
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。
Affine Super Yangian and Weyl groupoid
本文定义仿射超 Yangian 的极简与 Drinfeld 表示，证明二者同构，并构造不同单根系下的同构，进而用生成元刻画 Weyl 群胚及其作用。
A Drinfeld Presentation of the Queer Super-Yangian
本文通过块高斯分解和 RTT 关系，提出 queer 超 Yangian 的 Drinfeld 展开，并获得其中心系列的新表达式。
Representations of Super Yangians with Gelfand-Tsetlin bases
研究超杨代数的表示，通过评价同态将其协变张量模视为超杨代数模，并给出简单商模块的 tame 条件，扩展了纳扎罗夫和塔拉索夫的工作。
We apply novel techniques in planar superconformal Yang-Mills theory which stress the role of the Yangian algebra. We compute the first two Casimirs of the Yangian, which are identified with the first two local abelian Hamiltonians with periodic boundary conditions, and show that they annihilate the chiral primary states. We streamline the derivation of the R-matrix in a conventional spin model, and extend this computation to the gauge theory. We comment on higher-loop corrections and higher-loop integrability.
In order to extend the geometrization of Yangian $R$-matrices from Lie algebras $gl(n)$ to superalgebras $gl(M|N)$, we introduce new quiver-related varieties which are associated with representations of $gl(M|N)$. In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre tran…
Irreducibility of the tensor product of Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) evaluation modules
本文研究了超Yangian $ \mathrm{Y}(\mathfrak{gl}_{m|n}) $ 评估模块张量积的不可约性条件，基于Gelfand-Tsetlin基的存在性，给出了简单模块的判据。
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