R-matrix presentation for (super)-Yangians Y(g)

The $R$-matrix presentation for the Yangian of a simple Lie algebra

Starting from a finite-dimensional representation of the Yangian $Y(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$ in Drinfeld's original presentation, we construct a Hopf algebra $X_\mathcal{I}(\mathfrak{g})$, called the extended Yangian, whose defining relations are encoded in a ternary matrix relation built from a specific $R$-matrix $R(u)$. We prove that there is a surjective Hopf algebra morphism $X_\mathcal{I}(\mathfrak{g})\twoheadrightarrow Y(\mathfrak{g})$ whose kernel is generated as an ideal by the coefficients of a central matrix $\mathcal{Z}(u)$. When the underlying represe…

扭曲杨代数的Drinfeld型呈现

A Drinfeld type presentation of twisted Yangians of quasi-split type

本文提出了一类扭曲杨代数的Drinfeld型呈现，并展示了其与反射代数的同构关系。

$D$型twisted Yangian的新表示

New presentation of the twisted Yangian of type $D$

本文构造了具有有限个生成元的$D$型twisted Yangian新表示，并借此提出了$D$型twisted affine Yangian的新定义。

与李超代数 $\mathfrak{osp}(1|2m)$ 相关联的 Yangian 的高斯生成元

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。

BI、CI、DI型扭曲杨代数及Drinfeld型电流关系

Twisted Yangians of types BI, CI, DI and Drinfeld type current relations

研究BI、CI、DI型分裂对称对相关的扭曲杨代数，通过新的转置表示，在R - 矩阵表示内推导其Drinfeld型电流表示，建立了与Drinfeld表示的同构，还得到张量积分解、Poincaré - Birkhoff - Witt基及余理想余积。

Parabolic presentations of the Yangian Y(gl_n)

We derive some new presentations for the Yangian associated to the Lie algebra gl_n(C) that are adapted to parabolic subalgebras. At one extreme, the presentation is just the usual RTT presentation, whilst at the other extreme it is a variation on Drinfeld's presentation. All these presentations play an important role in our subsequent article "Shifted Yangians and finite W-algebras".

Parabolic presentations of the super Yangian $Y(gl_{M|N})$

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.

模超杨振子Y(M|N)的抛物型表示

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。

Equivalences between three presentations of orthogonal and symplectic Yangians

We prove the equivalence of two presentations of the Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ and we also show the equivalence with a third presentation when $\mathfrak{g}$ is either an orthogonal or a symplectic Lie algebra. As an application, we obtain an explicit correspondence between two versions of the classification theorem of finite-dimensional irreducible modules for orthogonal and symplectic Yangians.

Parabolic presentations of the super Yangian $Y(\mathfrak{gl}_{M|N})$ associated with arbitrary 01-sequences

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 $\mathfrak{gl}_{M|N}$.

A Drinfeld presentation for the twisted Yangian $Y_3^+$

We define the Drinfeld generators for $Y_3^+$, the twisted Yangian associated to the Lie algebra $\mathfrak{so}_3(\mathbb{C})$. This allows us to define shifted twisted Yangians, which are certain subalgebras of $Y_3^+$. We show that there are families of homomorphisms from the shifted twisted Yangians in $Y_3^+$ to the universal enveloping algebras of various orthogonal and symplectic Lie algebras, and we conjecture that the images of these homomorphisms are isomorphic to various finite $W$-algebras.

A型Yangian double的中心

Center of the Yangian double in type A

该研究证明A型Yangian double的R矩阵与Drinfeld表示同构，构造了临界水平下其完备形式的中心元素，借助R矩阵表示的PBW定理算出这些元素在Harish-Chandra型同态下的像，并验证其与Wakimoto模中中心元素的本征值一致。

From Twisted Quantum Loop Algebras to Twisted Yangians

We prove how the Yangian of $\mathfrak{gl}_N$ in its RTT presentation and Olshanski's twisted Yangians for the orthogonal and symplectic Lie algebras can be obtained by a degeneration process from the corresponding quantum loop algebra and some of its twisted analogues.

RTT presentation of finite W-algebras

We construct a wide class of finite W-algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow to construct the W-algebras as exchange algebras, the R-matrix being the Yangian's one. As an application, we classify all irreducible finite dimensionnal representations of these W-algebras and determine their center.

B型和C型Yangian的抛物型表示

Parabolic presentations of Yangian in types $B$ and $C$

本文通过对称合成参数化和块矩阵高斯分解，建立了B、C型扩展Yangian的抛物型表示，并由此得到其中心的新公式。

扭曲$q$-Yangian与仿射$\imath$量子群之间的同构：AI型

Isomorphism between twisted $q$-Yangians and affine $\imath$quantum groups: type AI

本研究利用高斯分解，建立了AI型对称对对应的扭曲$q$-Yangian与仿射$\imath$量子群的显式同构，并由此得到该类仿射$\imath$量子群的PBW型基。

例外李超代数 $D(2,1;λ)$ 的 Drinfeld 超杨代数

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。

queer 超 Yangian 的 Drinfeld 展开

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。

平移扭曲 Yangian 与经典型有限 $W$-代数

Shifted twisted Yangians and finite $W$-algebras of classical type

本文通过引入抛物型表示和截断平移扭曲 Yangian，建立其 PBW 基，并证明其同构于经典型有限 $W$-代数，从而为 B、C 型偶幂零元及 D 型双 Jordan 块幂零元提供表示。

Shifted Yangians and finite W-algebras

We give a presentation for the finite W-algebra associated to a nilpotent matrix inside the general linear Lie algebra over C. In the special case that the nilpotent matrix consists of n Jordan blocks each of the same size l, the presentation is that of the Yangian of level l associated to the Lie algebra gl_n, as was first observed by Ragoucy and Sorba. In the general case, we are lead to introduce some generalizations of the Yangian which we call the shifted Yangians.

Classification of Hopf superalgebra structures on Drinfeld super Yangians

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.

仿射超Yangian的余积与Weyl广群作用

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。

修改的Drinfeld-Cartan级数的余乘积用于类型A的Yangian和量子 affine 代数

Coproduct of modified Drinfeld-Cartan series for Yangians and quantum affine algebras in type A

本文研究了类型A的Yangian和特定类型A_2的量子 affine 代数的修改Drinfeld-Cartan生成级数的余乘积公式，并给出了正预基本表示的显式描述。

W-superalgebras as truncation of super-Yangians

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.

拟分裂 iYangian：极简表示与余理想结构

Quasi-split iYangians: minimalistic presentations and coideal structures

本文研究拟分裂对称对关联的 iYangian，给出极简表示，构造到 Yangian 的嵌入并识别余理想，证明表示同构，并推导电流三角估计以分析限制权。

扭曲的超Yangian代数的准分裂型A

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。

Super Yangian Y(osp(1|2)) and the Universal R-matrix of its Quantum Double

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)).

Gauss Decomposition of the Yangian Y(gl_{m|n})

We describe a Gauss decomposition for the Yangian Y(gl_{m|n}) of the general linear Lie superalgebra. This gives a connection between this Yangian and the Yangian of the classical Lie superalgebra Y(A(m-1,n-1)) (with m and n not equal) defined and studied in papers by Stukopin, and suggests natural definitions for the Yangians Y(sl_{n|n}) and Y(A(n,n)). We also show that the coefficients of the quantum Berezinian generate the centre of Y(gl_{m|n}). This was conjectured by Nazarov in 1991.

仿射超 Yangian 与非矩形 W-超代数

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。

交换的仿射超杨代数边收缩子代数：源自边收缩

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。

Representations of Quantum Affine Algebras in their $R$-Matrix Realization

We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.

Classification of reflection matrices related to (super) Yangians and application to open spin chain models

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.

仿射超 Yangian 与 Weyl 群胚

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。

Lax pair and super-Yangian symmetry of the non-linear super-Schrödinger equation

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…

A型和C型超Yangian的表示

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。

Twisted super Yangians of type AIII 及其表示

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。

$B(n,m)$型超Yangian的表示

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。

正交辛超杨氏代数

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。

与李超代数 ${\ rak{osp}}(1|2n)$ 相关的 Yangian 的表示

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。

D(n,m)型超Yangian的表示

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。

从量子loop超代数到超Yangian

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。

双重Yang代数与量子顶点代数，I

Double Yangians and quantum vertex algebras, I

本文引入了双重Yang代数的广义Cartan矩阵的中心扩展，并构造了新的场呈现形式，证明了真空模的量子顶点代数结构，并建立了模范畴的等价关系。

Isomorphism between super Yangian and quantum loop superalgebra. I

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 $U_{\hbar}(LA(m,n))$.

Quantum Double of Yangian of Lie Superalgebra $A(m,n)$ and computation of Universal $R$-matrix

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.

Finite W-superalgebras and truncated super Yangians

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.

Representations of the orthosymplectic Yangian

We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $\frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, they are classified in terms of highest weights. We give explicit constructions of a family of elementary modules of the Yangian and show that a wide class of irreducible representations can be produced by taking tensor products of the elementary modules.

李超代数 $B(m,n)$ 的 Drinfeld 超 Yangian 的 Hopf 超代数结构

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。

Super Yangian Double $DY( gl(1|1))$ and Its Gauss Decomposition

We extend Yangian double to super (or graded) case and give its Drinfel'd generators realization by Gauss decomposition.

扭超 Yangian 的量子 Berezinian

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 Yangian Double DY(gl(1|1)) With Center Extension

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

Super Yangian Double and its Central Extension

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.

超 Yangian 的 Jacobi-Trudi 恒等式与 Drinfeld 函子

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。

特征2上的超Yangian

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。

Finite W-superalgebras via super Yangians

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.