N. J. MacKay
We describe recent work on the twisted Yangians Y(g,h) which arise as boundary remnants of Yangians Y(g) in 1+1D integrable field theories, bringing out the special role played by the requirement that (g,h) be a symmetric pair.
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We describe the twisted Yangians Y(g,h) which arise as boundary remnants of Yangians Y(g) in 1+1D integrable field theories. We describe and extend our recent construction of the intertwiners of their representations (the rational boundary S- or 'K'-matrices) and perform a case-by-case analysis for all pairs (g,h), giving the h-decomposition of Y(g,h)-representations where possible.
We show that, in the open Hubbard model with integrable boundary conditions, the bulk Yangian symmetry is broken to a twisted Yangian. We prove that the associated charges commute with the Hamiltonian and the reflection matrix, and that they form a coideal subalgebra.
We give the principal realization of the twisted Yangians of orthogonal and symplectic types. The new bases are interpreted in terms of discrete Fourier transform over the cyclic group Z_N.
We study quantized enveloping algebras called twisted Yangians. They are analogues of the Yangian Y(gl(N)) for the classical Lie algebras of B, C, and D series. The twisted Yangians are subalgebras in Y(gl(N)) and coideals with respect to the coproduct in Y(gl(N)). We give a complete description of their finite-dimensional irreducible representations. Every such representation is highest weight and we give necessary and sufficient conditions for an irreducible highest weight representation to be finite-dimensional. The result is analogous to Drinfeld's theorem for the ordinary Yangians. Its de…
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。
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.
We define the twisted affine Yangian of type $C$ and construct surjective homomorphisms from twisted affine Yangians of type $C$ to the universal enveloping algebra of the rectangular $W$-algebra associated with $\mathfrak{so}(ln)$ and a nilpotent element whose Jordan form corresponds to the partition $(l^n)$ in the case when $l$ and $n$ are even.
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.
With any skew Young diagram one can associate a one parameter family of "elementary" modules over the Yangian $\Yg(\gł_N)$. Consider the twisted Yangian $\Yg(\g_N)\subset \Yg(\gł_N)$ associated with a classical matrix Lie algebra $\g_N\subset\gł_N$. Regard the tensor product of elementary Yangian modules as a module over $\Yg(\g_N)$ by restriction. We prove its irreducibility for generic values of the parameters.
We show that the truncation of twisted Yangians are isomorphic to finite W-algebras based on orthogonal or symplectic algebras. This isomorphism allows us to classify all the finite dimensional irreducible representations of the quoted W-algebras. We also give an R-matrix for these W-algebras, and determine their center.
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 initiate a theory of highest weight representations for twisted Yangians of types B, C, D and we classify the finite-dimensional irreducible representations of twisted Yangians associated to symmetric pairs of types CI, DIII and BCD0.
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.
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。
Twisted q-Yangians and Sklyanin determinants
本文通过将量子仿射代数表示为两个 q-Yangian 副本的乘积，研究其及扭曲版本的不变量理论，并引入扭曲 Sklyanin 行列式、建立相关恒等式。
We study the convolution algebra $H^{G\times \CC^{*}}_{*}(Z)$ of $G$-equivariant homology group on the Steinberg variety of type B/C and define an algebra $\widetilde{Y}$ that maps to $H^{G\times \CC^{*}}_{*}(Z)$. The Drinfeld new realization of the twisted Yangian associated to symmetric pairs is a quotient of $\widetilde{Y}$. We also study the $G$-equivariant case and prove that the twisted Yangian is the deformation of the twisted current algebra.
Twisted Yangians and Steinberg varieties of type C
研究型C广义steinberg变种的equivariant同调，并证明扭转杨代数AIII_{2n}^{(τ)}到它的代数同态映射
We study analogues of the Yangian of the Lie algebra $gl_N$ for the other classical Lie algebras $so_N$ and $sp_N$. We call them twisted Yangians. They are coideal subalgebras in the Yangian $Y(gl_N)$ of $gl_N$ and admit homomorphisms onto the universal enveloping algebras $U(so_N)$ and $U(sp_N)$ respectively. In every twisted Yangian we construct a family of maximal commutative subalgebras parametrized by the regular semisimple elements of the corresponding classical Lie algebra. The images in $U(so_N)$ and $U(sp_N)$ of these subalgebras are also maximal commutative.
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 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 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.
We construct an explicit set of generators for the finite W-algebras associated to nilpotent matrices in the symplectic or orthogonal Lie algebras whose Jordan blocks are all of the same size. We use these generators to show that such finite W-algebras are quotients of twisted Yangians.
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。
We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.
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…
Analogs of the classical Sylvester theorem have been known for matrices with entries in noncommutative algebras including the quantized algebra of functions on GL(N) and the Yangian for gl(N). We prove a version of this theorem for the twisted Yangians Y(g(N)) associated with the orthogonal and symplectic Lie algebras g(N)=o(N) or sp(N). This gives rise to representations of the twisted Yangian Y(g(N-M)) on the space of g(M)-homomorphisms Hom(W,V), where W and V are finite-dimensional irreducible modules over g(M) and g(N), respectively. In the symplectic case these representations turn out to…
Based on the (quantum) twisted Yangians, integrable systems with special boundary conditions, called soliton non-preserving (SNP), may be constructed. In the present article we focus on the study of subalgebras of the (quantum) twisted Yangians, and we show that such a subalgebra provides an exact symmetry of the rational transfer matrix. We discuss how the spectrum of a generic transfer matrix may be obtained by focusing only on two types of special boundaries. It is also shown that the subalgebras, emerging from the asymptotics of tensor product representations of the (quantum) twisted Yangi…
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 twisted Yangians of types B, C, D: II
本文继续研究B、C、D型对称对相关的扭曲Yangian的有限维不可约表示，证明了最高权满足特定关系，并分类了两种配对情形下的表示。
Associating deformed $ϕ$-coordinated modules for the quantum affine vertex algebra with orthogonal twisted $h$-Yangians
本文通过引入变形 $ϕ$-协调模的概念，将量子 affine 顶点代数与正交扭曲 $h$-Yangians 关联，并展示其应用。
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.
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。
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…
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。
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。
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。
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。
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。
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。
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。
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。
Invariants of the extended twisted $h$-Yangian
研究扩展扭曲\(h\)-杨式代数\({\rm Y}_{N,h}^{\text{tw}}\)，通过展示其具有特定代数的限制模和拟模结构，利用量子Feigin - Frenkel中心元素得出中心元素、不变量公式及交换族等贡献。
We use the theory of reductive dual pairs due to Howe to obtain explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra, and of the twisted Yangians corresponding to the symplectic and orthogonal Lie algebras.
A note on constructing quasi modules for quantum vertex algebras from twisted Yangians
本文通过双 Yangian 的子代数结构，构造量子仿射顶点代数的拟模，并给出相关中心元素与真空模不变量的显式公式。
For each of the classical Lie algebras $g(n)=o(2n+1), sp(2n), o(2n)$ of type B, C, D we consider the centralizer of the subalgebra $g(n-m)$ in the universal enveloping algebra $U(g(n))$. We show that the $n$th centralizer algebra can be naturally projected onto the $(n-1)$th one, so that one can form the projective limit of the centralizer algebras as $n\to\infty$ with $m$ fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by $A_m$. We explicitly construct an algebra isomorphism $A_m=Z\otimes Y_m$, where $Z$ is a commutative alge…
Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in Yanigian are discussed. As a corollary, similar relations are deduced for shifted Schur functions.
Similarity transformations and eigenvalue relations of monodromy operators composed of Jordan-Schwinger type L matrices are considered and used to define Yangian symmetric correlators of n-dimensional theories. Explicit expressions are obtained and relations are formulated. In this way basic notions of the Quantum inverse scattering method provide a convenient formulation for high symmetry and integrability not only in lower dimensions.
Twisted Hopf algebra $sl_ξ(2)$ gives rise to a deformation of the Yangian ${\cal Y}(sl(2))$. The corresponding deformations of the integrable XXX-spin chain and the Gaudin model are discussed.
Correlators based on $s\ell_2$ Yangian symmetry and its quantum deformation are studied. Symmetric integral operators can be defined with such correlators as kernels. Yang-Baxter operators can be represented in this way. Particular Yangian symmetric correlators are related to the kernels of QCD parton evolution. The solution of the eigenvalue problem of Yangian symmetric operators is described.
Shifted twisted Yangians of quasi-split ADE types
本文引入了偏移i阳离子代数，并展示了其与有限W代数的同构关系，探讨了其在仿射格拉斯曼切片固定点区域中的应用。
This is a review paper on the algebraic structure and representations of the A type Yangian and the B, C, D types twisted Yangians. Some applications to constructions of Casimir elements and characteristic identities for the corresponding Lie algebras are also discussed.
Let $G_M$ be one of the complex Lie groups $O_M$ and $Sp_M$. The irreducible finite-dimensional representations of the group $G_M$ are labeled by partitions $μ$ satisfying certain extra conditions. Let $U$ be the representation of $G_M$ corresponding to $μ$. Regard the direct product $G_N\times G_M$ as a subgroup of $G_{N+M}$. Let $V$ be the irreducible representation of $G_{N+M}$ corresponding to a partition $λ$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a natural action of the group $G_N$. Let $n$ be sum of parts of $λ$ less the sum of parts of $μ$. For any choice of a stan…
The decompositions of the Fock and Basic modules of the affine Lie algebra \hat{sl(N)} into irreducible submodules of the Yangian algebra Y(gl(N)) are constructed. Each of the irreducible submodules admits the unique up to normalization eigenbasis of the maximal commutative subalgebra of the Yangian. The elements of this eigenbasis are identified with specializations of Macdonald symmetric functions where both parameters of these functions approach an N-th primitive root of unity.
In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.
Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs
本文研究经典群在ALE空间上的瞬子模空间（即箭图簇上对合的不动点集），通过稳定包络构造扭曲Yangian表示，计算K-矩阵实例并明确良定义条件。
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…
A general scheme of construction of Drinfeldians and Yangians from quantum non-twisted affine Kac-Moody algebras is presented. Explicit description of Drinfeldians and Yangians for all Lie algebras of the classical series A, B, C, D are given in terms of a Cavalley basis.
The localized equivariant homology of the quiver variety of type $A_{N-1}^{(1)}$ can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack($\mathfrak{gl}_N$) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for $\mathfrak{sl}_N$ and the affine Lie algebra $\hat{\mathfrak{sl}}_N$, on both sides. Consequently we obtain affine Yangian action on the Fock space.
By using the non-symmetric Hermite polynomials and a technique based on the Yangian Gelfand-Zetlin bases, we decompose the space of states of the Calogero model with spin into irreducible Yangian modules, construct an orthogonal basis of eigenvectors and derive product-type formulas for norms of these eigenvectors.
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