Drinfeld Functor and Finite-Dimensional Representations of Yangian
Tomoyuki Arakawa
本文将Drinfeld关于Drinfeld函子的结果推广到l>n的情形。作为结果，我们给出了Yangian Y（sl_n）的有限维表示用Kazhdan-Lusztig多项式表示的性质.
We extend the results of Drinfeld on Drinfeld functor to the case l>n. We present the character of finite-dimensional representations of the Yangian Y(sl_n) in terms of the Kazhdan-Lusztig polynomials as a consequence.
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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.
Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case Lie superalgebra $sq_1$ that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev-Reshetikhin-Takhtadzhjan approach, are isomorphic.
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。
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 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 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。
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.
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 …
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。
Twisted Yangians of split 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。
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。
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。
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。
Integral forms for Yangians of classical type
本文为经典类型Yangian构造了由Drinfeld生成元分次幂生成的Z[1/2]-积分形式，建立三角分解与PBW基，并在特征p>3时将其模p约化与受限Yangian等同。
A construction of Hopf algebra cocycles for the double Yangian $DY(SL_{2})$
我们构造了一个Hopf代数上循环的杨双$DY(SL_{2})$，共轭Drinfeld的余积到通常的。为了做到这一点，我们因式分解Drinfeld的余积的两个“相反”版本之间的扭曲，在早期的工作中引入V. Rubtsov和第一作者，使用代数的分解在其负和非负模式子代数。
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。
On Drinfeld's second realization of the AdS/CFT su(2|2) Yangian
我们在psu（2）的基础上构造了Drinfeld的Yangian的第二种实现|2）xR^3对称性。第二种实现传统上更适合于导出量子二重和相对于第一种实现的泛R矩阵，最初由Beisert获得，并且它通常更有用，以便研究有限维表示。我们证明了这两种实现是同构的，其中同构几乎是Drinfeld给出的简单李代数的标准同构，但需要一些关键的修正来解释中心荷。我们还评估了第二实现的基本表示的发电机，发现有趣的结果，一些发电机的快度变量得到提升的能量特征值。
Drinfeld realization of the centrally extended $\mathfrak{psl}(2|2)$ Yangian algebra with the manifest coproducts
本文给出了中心扩展李超代数 $\mathfrak{psl}(2|2)$ 的 Yangian 代数的 Drinfeld 实现，并证明了其 Hopf 代数结构及显式余积的存在性。
Compatibility between truncation and coproducts for quantum affine algebra and Yangian of $\mathfrak{sl}_2(\mathbb{C})$
本文证明$\mathfrak{sl}_2(\mathbb{C})$的Yangian和量子仿射代数的标准余积与截断商相容，并给出截断级数的余积公式。
Super Yangian Double DY(gl(1|1)) With Center Extension
利用超RS构造法和Gauss分解，得到了中心扩张的超Yangian双$DY(gl(1|1))$的Drinfeld电流实现
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。
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。
Duality Theorem and Hom Functor in Braided Tensor Categories
将Blatter-Montgomery对偶定理推广到辫张量范畴。证明了对于任意两个辫子Yetter-Drinfeld模$V$和$W$，$Hom(V,W)$是辫子Yetter-Drinfeld模.
We introduce an analogue of the composition of the Cherednik and Drinfeld functor for twisted Yangians. Our definition is based on the Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We introduce a skew analogue of the composition of the Cherednik and Drinfeld functors for twisted Yangians. Our definition is based on the skew Howe duality, and originates from the centralizer construction of twisted Yangians due to Olshanski. Using our functor, we establish a correspondence between intertwining operators on the tensor products of certain modules over twisted Yangians, and the extremal cocycle on the hyperoctahedral group.
We study a composition of two functors. The first one, from the category of modules over the Lie algebra $\gl_m$ to the category of modules over the degenerate affine Hecke algebra of $GL_N$, was introduced by I. Cherednik. The second functor is a skew version of the functor from the latter category to the category of modules over the Yangian $\Y(\gl_n)$ due to V. Drinfeld. We give a representation theoretic explanation of a link between intertwining operators on tensor products of $\Y(\gl_n)$-modules, and the `extremal cocycle' on the Weyl group of $\gl_m$ introduced by D. Zhelobenko. We esta…
The isomorphism between Drinfeld's new realization and the FRT realization is proved for the Yangian algebra Y(so_3) by using Gauss decomposition.
The Yangian double $\text{DY}_{\hbar}(\mathfrak{g}_N)$ is introduced for the classical types of $\mathfrak{g}_N=\mathfrak{o}_{2n+1}$, $\mathfrak{sp}_{2n}$, $\mathfrak{o}_{2n}$. Via the Gauss decomposition of the generator matrix, the Yangian double is given the Drinfeld presentation. In addition, bosonization of level $1$ realizations for the Yangian double $\text{DY}_{\hbar}(\mathfrak{g}_N)$ of non-simply-laced types are explicitly constructed.
Universal R-matrix Of The Super Yangian Double DY(gl(1|1))
基于Drinfeld实现的超Yangian双DY（gl（1| 1）），给出了它的配对关系和泛R-矩阵。利用泛R-矩阵的求值表示，给出了DY（gl（1））的另一种实现$L^{\pm}(u)$|（1）得到。这两个实现DY（gl（1|（1））是由Ding-Frenkel映射的超对称扩张联系起来的。
We study the composition of the functor from the category of modules over the Lie algebra gl_m to the category of modules over the degenerate affine Hecke algebra of GL_N introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian Y(gl_n) due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the tensor products of Y(gl_n)-modules, and the `extremal cocycle' on the Weyl group of gl_m defined by D. Zhelobenko. We also establish a connection between the composition of two functors…
Representations of Yangians with Gelfand-Zetlin Bases
研究了Yangian $Y(gl_N)$上的一类有限维模.代数$Y(gl_N)$配备了一个由链$Y(gl_1)\subset Y(gl_2)\subset...\subset Y(gl_N)$中所有代数的中心生成的杰出的极大交换子代数$A(gl_n)$。研究了有限维$A(gl_N)$-模的半单作用。我们称这些模块为Tame。我们给出了不可约Tame模的Drinfeld多项式的一个刻画。我们证明了每个不可约的驯服模分裂成一个张量积的模块对应的斜杨图和一些一维模。$A(gl_N)$在不可约模中的特征基称为Gelfand-Zetlin基。给出了代数$Y(gl_N)$的Drinfeld生成元对Gelfand-Zetlin基向量的作用的显式公式。
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。
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…
In this work, using the functoriality of Drinfeld center of fusion categories, we generalize an earlier result on the functoriality of full center of simple separable algebras in a fixed fusion category to all fusion categories. This generalization produces a new center functor, which involves both Drinfeld center and full center and will be called the pointed Drinfeld center functor. We prove that this pointed Drinfeld center functor is a symmetric monoidal equivalence. It turns out that this functor provides a precise and rather complete mathematical formulation of the boundary-bulk relation…
Weyl group extension of quantized current algebras
在本文中，我们在几个方向上扩展量子仿射代数$U_q(\hat {\gg})$和Yangians的Drinfeld电流实现：我们构造了${\gg}$的非单根的电流算子，根据当前算子定义一个新的辫群作用，并描述相应“”的泛R-矩阵。关于流算子的无穷乘积形式和某些积分形式的Drinfeld "余乘。
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。
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.
Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers
本研究给出强幺半函子的双伴随函子成为Frobenius幺半函子的一般条件，进一步得到其可提升为Drinfeld中心上辫子Frobenius幺半函子的更强条件，并通过Hopf代数态射构造了具体实例。
On the R-matrix realization of Yangians and their representations
本文研究了B、C、D型单李代数a的Yangians Y（a）。代数Y（a）可以看作是广义杨数X（a）的商，其定义关系用R-矩阵的形式表示。本文研究了代数X（a）的代数结构和表示。我们证明了X（a）的Poincare-Birkhoff-Witt定理的一个类似定理，并证明了Yangian Y（a）可以实现为X（a）的一个子代数.进一步，我们给出了X（a）的有限维不可约表示的分类定理的独立证明，它蕴涵了Yangians Y（a）的Drinfeld相应定理.我们也给出了明确的建设的所有基本代表性的Yangians。
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…
Let ${\mathfrak g}$ be a complex semisimple Lie algebra, and $Y_h({\mathfrak g})$, $U_q(L{\mathfrak g})$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q=\exp(πi h)$. When $h$ is not a rational number, we constructed in arXiv:1310.7318 a faithful functor $Γ$ from the category of finite-dimensional representations of $Y_h ({\mathfrak g})$ to those of $U_q(L{\mathfrak g})$. The functor $Γ$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of $Y_h({\mathfrak…
Skew representations of twisted Yangians
经典西尔维斯特定理的类似物已经被称为非交换代数中的矩阵，包括GL（N）上的量子化函数代数和gl（N）的Yangian。我们证明了这个定理的一个版本的扭曲Yangians Y（g（N））与正交和辛李代数g（N）=o（N）或sp（N）。这就产生了扭曲杨格Y（g（N-M））在g（M）-同态空间Hom（W，V）上的表示，其中W和V分别是g（M）和g（N）上的有限维不可约模。在辛的情况下，这些表示原来是不可约的，我们确定它们通过计算相应的Drinfeld多项式。我们还应用量子Sylvester定理实现了扭曲Yangian作为泛包络代数中某些中心化子的投射极限。
Finite-dimensional irreducible representations of twisted Yangians
我们研究量子化包络代数称为扭曲杨格。它们类似于B、C和D级数的经典李代数的Yangian Y（gl（N））。扭杨格数是Y（gl（N））中的子代数和关于Y（gl（N））中的余积的余理想。我们给出了它们的有限维不可约表示的完整描述。每一个这样的表示是最高权，我们给出了必要和充分条件，不可约的最高权表示是有限维的。结果类似于德林费尔德定理的普通杨格尔。它的详细证明的A系列也转载。对于最简单的扭曲杨氏，我们构造一个明确的实现，为每个有限维不可约表示在张量积的表示相应的李代数。
Coinvariants for Yangian doubles and quantum KZ equations
我们提出了一个量子版本的建设的KZ系统的方程作为一个平坦的连接上的空间的共不变量的表示卡茨-穆迪代数的张量积。在这里，我们考虑表示的张量积的杨双和计算的变形的子代数所产生的正规函数的有理曲线与标记点的共不变量。我们观察到Drinfeld量子Casimir元可以看作是Yangian二重中Sugawara张量零模的变形。这些成分用于定义一个相容的差分方程系统，我们将其与I. Frenkel和N.列谢提欣
GKLO representations of twisted Yangians in type $\mathsf{AI}$ and quantizations of symmetric quotients of the affine Grassmannian
本文构造了类型AI扭曲Yangian的GKLO表示，定义其截断，并证明截断的扭曲Yangian量子化仿射Grassmannian横截切片商的概形。
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。
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。
On a class of representations of the Yangian and moduli space of monopoles
构造了一类新的Yangians $Y(\frak{g})$和$Y(\frak{b})$的无穷维表示，它们对应于复半单代数$\frak{g}$及其Borel子代数$\frak{b}\subset\frak{g}$.它是基于推广的Drinfeld实现的$Y(\frak{g})$，$\frak{g}=\frak{gl}(N)$的量子子项的情况下，任意半单李代数$\frak{g}$。描述了与构造的表示相关联的泊松几何。特别是它示出的基本辛叶同构的模空间的$G$ -单极定义为组件的空间的基本映射的$\mathbb{P}^1$到广义旗流形$X=G/B$。因此，所构造的Yangian表示可以被认为是单极子的模空间的量子化。
The Lyubashenko Modular Functor for Drinfeld Centers via Non-Semisimple String-Nets
本文证明了由枢轴有限张量范畴的投射对象构建的有限余完备弦网模函子与由Drinfeld中心构建的Lyubashenko模函子等价，推广了Levin-Wen弦网的结果到非半单情形。
We study a functor from anti-Yetter Drinfeld modules to contramodules in the case of a Hopf algebra $H$. Some byproducts of this investigation are the establishment of sufficient conditions for this functor to be an equivalence, verification that the center of the opposite category of $H$-comodules is equivalent to anti-Yetter Drinfeld modules, and the observation of two types of periodicities of the generalized Yetter-Drinfeld modules introduced previously. Finally, we give an example of a symmetric $2$-contratrace on $H$-comodules that does not arise from an anti-Yetter Drinfeld module.
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。
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。
For M and N finite module categories over a finite tensor category C, the category Rex_C(M,N) of right exact module functors is a finite module category over the Drinfeld center Z(C). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map C-C-bimodule functors to objects of Z(C), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if M and N are exact C-modules and C is pivotal, then the Z(C)-module Rex_C(M,N) is exact. We compute its relative…
We study the Yangian of the sl(2|1) Lie superalgebra in a multi-parametric four-dimensional representation. We use Drinfeld's second realization to independently rederive the R-matrix, and to obtain the antiparticle representation, the crossing and the unitarity condition. We consistently apply the Yangian antipode and its inverse to the individual particles involved in the scattering. We explicitly find a scalar factor solving the crossing and unitarity conditions, and study the analytic structure of the resulting dressed R-matrix. The formulas we obtain bear some similarities with those fami…
Universal blocks of the AdS/CFT Scattering Matrix
我们确定某些块的S-矩阵描述的散射的束缚态的AdS 5 × S5超弦，允许在通用的R-矩阵的杨双表示。对于这些情况下，我们使用的公式Drinfeld的第二次实现的Yangian在任意的束缚态表示，以获得相应的R-矩阵的显式表达式。然后，我们表明，这些表达式与先前获得的S-矩阵块完全匹配。
Duality for action bialgebroids
研究线性对偶及量子对偶原理下Drinfeld函子对作用双代数胚的影响，证明辫子交换Yetter-Drinfeld代数在对偶双代数胚上的性质及作用双代数胚构造与对偶/量子对偶函子的交换性。
Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
本文通过有向图方法构造了A型Dynkin图相关的Yang代数及其矩形表示，并展示了其与Gelfand-Tsetlin基的关联。
Projection formulas and induced functors on centers of monoidal categories
本文研究了单子adjunction诱导的braided lax单子函子，通过投射公式 morphisms 的可逆性，探讨了Hopf代数的限制与诱导沿态射的单子adjunction，并应用这些函子到 affine代数群、量子群和Radford-Majid双积Hopf代数的例子中。
The meromorphic R-matrix of the Yangian
本文证明 Yangian 的有限维表示上不存在有理交换约束，转而构造两个亚纯交换约束，其渐近展开与 Drinfeld 泛 R 矩阵一致，从而给出泛 R 矩阵存在性的构造性证明。
Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra $\hat g$. In case $g$ is the symplectic Lie algebra $sp_N$, we introduce an affine, reduced, irreducible, normal quiver variety $Z$ which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on the zastava space $Z$ can be described in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient o…
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