Jonathan Brundan, Alexander Kleshchev
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".
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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。
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.
We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular $W$-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that parabolic inductions for representations of the rectangular $W$-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super setting.
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。
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。
A Drinfeld type presentation of twisted Yangians
本文通过Gauss分解方法为AI型扭曲Yangian建立了Drinfeld型电流表示，解决了长期未解问题，并有望推广至其他扭曲Yangian。
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。
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.
It is well-known that the Gauss decomposition of the generator matrix in the $R$-matrix presentation of the Yangian in type $A$ yields generators of its Drinfeld presentation. Defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations. It has been an open problem since the pioneering work of Drinfeld to extend this result to the remaining types. We give a solution for the classical types $B$, $C$ and $D$ by constructing an explicit isomorphism between the $R$-matrix and Drinfeld presentations of the Yangian. It is based on …
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-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 give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n) and osp(m|2n).
Minimalistic Presentation and Coideal Structure of Twisted Yangians
本文通过Drinfeld型表示法引入扭曲杨代数的极简表示，并证明其作为杨代数的右余子代数结构，同时描述其在J表示下的同构性。
Affine $\imath$quantum groups and twisted Yangians in Drinfeld presentations
本文通过仿射 ı 量子群 Drinfeld 呈现的退化构造了分裂型扭 Yangian，证明其具有 PBW 型基且为扭流代数的形变，并猜想其与 RTT 呈现中的对应代数同构。
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…
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。
We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.
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。
Given an affine Kac-Moody algebra, we explain how to construct a coproduct for its associated Yangian. In order to prove that this coproduct is an algebra homomorphism, we obtain, in the first half of this paper, a minimalistic presentation of the Yangian when the Kac-Moody algebra is, more generally, symmetrizable.
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。
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.
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。
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。
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.
The twisted q-Yangians are coideal subalgebras of the quantum affine algebra associated with gl(N). We prove a classification theorem for finite-dimensional irreducible representations of the twisted q-Yangians associated with the symplectic Lie algebras sp(2n). The representations are parameterized by their highest weights or by their Drinfeld polynomials. In the simplest case of sp(2) we give an explicit description of all the representations as tensor products of evaluation modules. We prove analogues of the Poincare-Birkhoff-Witt theorem for the quantum affine algebra and for the twisted q…
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。
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。
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.
Shifted twisted Yangians and Slodowy slices in classical Lie algebras
本文引入 AI 型移位扭 Yangian，证明其半经典极限与 gl_n 半经典移位 Yangian 的 Dirac 约化一致，并由此得到 B、C、D 型非矩形幂零元 Slodowy 切片的截断同构、抛物型表示及偶幂零元情形的泊松表示。
The coproduct for the affine Yangian and the parabolic induction for non-rectangular $W$-algebras
利用仿射Yangian的余积与求值映射及非矩形W-代数的Miura映射，构造仿射Yangian到非矩形W-代数包络代数的同态，证明余积与抛物诱导相容，并刻画其像在仿射余集内的特殊情形。
The Yangian characters (or q-characters) are known to be closely related to the classical W-algebras and to the centers of the affine vertex algebras at the critical level. We make this relationship more explicit by producing families of generators of the W-algebras from the characters of the Kirillov-Reshetikhin modules associated with multiples of the first fundamental weight in types B and D and of the fundamental modules in type C. We also give an independent derivation of the character formulas for these representations in the context of the RTT presentation of the Yangians. In all cases …
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。
The Yangian of the Lie algebra sl_n is known to have different presentations, in particular the RTT realisation and the Drinfel'd realisation. Using the isomorphism between them, the explicit expressions of the comultiplication, the antipode and the counit in the Drinfel'd realisation of the Yangian Y(sl_n) are given. As examples, the cases of Y(sl_2) and Y(sl_3) are worked out.
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 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.
Twisted Yangians of small rank
本文研究小秩下CI、BDI和DIII型对称对的扭杨振宁代数，建立了与Olshanskii扭杨振宁代数及Molev-Ragoucy反射代数的同构，并构造了与Drinfeld表述的同构。
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。
We show that the Yangian Yn over gl_n possesses some features of the ring of regular functions on GL_n. In particular, we use the theory of quasideterminants to construct noncommutative flags associated to Yn. In so doing, a class of comodule algebras Fl(γ) for Yn (viewed as C[GL_n]) is revealed. As in the classic case, these flag algebras are related to the irreducible representations of Yn (viewed as U(gl_n)). In the course of defining the rings Fl(γ), connections to the new parabolic presentations of Yn given by Brundan and Kleshchev (2005) are uncovered.
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。
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。
Intertwiners of Yangian representations
本文构造了嵌入 Heisenberg 代数的 $g\ell(n)$ 型 Yangian 代数的 $N$ 阶赋值及其最高权表示，并显式构造了由满足 Yang-Baxter 关系的 $R$ 算子导出的缠绕算子，利用置换系数区分表示类型。
We study the truncated shifted Yangian $Y_{n,l}(σ)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent element $e\in \mathfrak{g} = \mathfrak{gl}_N(\mathbb{k})$. We obtain an explicit description of the centre of $Y_{n,l}(σ)$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(σ)$ to be the quotient of $Y_{n,l}(σ)$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main th…
Brundan and Kleshchev recently introduced a new family of presentations of the Yangian Y(gl_n) associated to the general linear Lie algebra gl_n, and thus provided a fresh approach to its study. In this article, we show how some of their ideas can be fruitfully extended to consider the Yangian Y(gl_{m|n}) associated to the Lie superalgebra gl_{m|n}. In particular, we give a new proof of the result by Nazarov that the quantum Berezinian is central.
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。
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。
Shuffle algebra realizations for modular Yangians
本文研究了在特征p>2的域上，sl_n的杨代数正子代数Y_n^(>)(k)的洗牌代数实现，确定了其核和像，并揭示了与特征p相关的轮条件。
The centralizer construction and Yangian-type algebras
本文通过推广中心化子构造得到一类新的Yangian型代数族，并证明其保留二次-线性定义关系等Yangian性质，同时为自由结合代数上的特殊双重Poisson括号提供量子化。
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。
On the uniqueness of Yangians
本文证明了单李代数多项式当前代数的齐次量子化至多唯一的断言，结合上同调与计算方法，输出了此类量子化的生成元与约化关系展示。
Parabolic quantum affine algebras
本文将非扭仿射Kac-Moody代数极大抛物子代数的等变映射代数实现推广到非极大情形，引入抛物量子仿射代数，构建PBW型基等结构，并在无重复条件下用带附加数据的Drinfeld多项式分类其有限维不可约表示。
We propose a new approach to study coideal algebras. It is well-known that Manin triples (or equivalently Lie bi-algebra structures) are the requirement to deform Lie algebras and to obtain quantum groups. In this paper, introducing some particular automorphisms of Manin triples, we define new structures that we call Lie bi-ideal structures. A link with coisotropic subalgebras is explained. We show that their deformation provide coideal algebras. As examples, we recover from our general construction the twisted Yangians, the q-Onsager algebra and the augmented q-Onsager algebra. As an importan…
We prove the equivalence of two presentations of deformed double current algebras associated to a complex simple Lie algebra, the first one obtained via a degeneration of affine Yangians while the other one naturally appeared in the construction of the elliptic Casimir connection. We also construct a specific central element of these algebras and, in type A, show that they contain a very large center for certain values of their parameters.
In this article, we deal with the structure of the spherical Hall algebra of coherent sheaves with parabolic structures on a smooth projective curve of arbitrary genus. We provide a shuffle-like presentation of the vector bundle part and show the existence of the generic form. We also prove that the spherical Hall algebra contains the characteristic functions on all the Harder-Narasimhan strata. These results together imply Schiffmann's theorem on the existence of Kac polynomials for quasi-parabolic vector bundles of fixed rank and multi-degree over the curve. On the other hand, the shuffle st…
Let $P_J$ be the standard parabolic subgroup of $SL_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J = L_JU_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue $θ: {\mathcal O}_q(P_J) \to{\mathcal O}_q(L_J) \otimes {\mathcal O}_q(P_J)$ of an induced coaction and the corresponding subalgebra ${\mathcal O}_q(P_J)^{\operatorname{co} θ} \subseteq {\mathcal O}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal O}_q(L_J)\# {\mathcal O}_q(P_J)^{\operatorname{co} θ}$ is isomorphic to ${\mathcal…
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