J. Donin, P. P. Kulish, A. I. Mudrov
For a given quasitriangular Hopf algebra $\Ha$ we study relations between the braided group $\tilde \Ha^*$ and Drinfeld's twist. We show that the braided bialgebra structure of $\tilde \Ha^*$ is naturally described by means of twisted tensor powers of $\Ha$ and their module algebras. We introduce universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices
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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.
Universal matrix Capelli identity
本研究提出泛型矩阵Capelli恒等式，阐释了推导反射方程代数与泛包络代数U(gl_(M|N))中所有量子immanant的Capelli恒等式的方法。
We construct a dynamical reflection equation algebra, $\tilde {\mathcal{K}}$, via a dynamical twist of the ordinary reflection equation algebra. A dynamical version of the reflection equation is deduced as a corollary. We show that $\tilde {\mathcal{K}}$ is a right comodule algebra over a dynamical analog of the Faddeev-Reshetikhin-Takhtajan algebra equipped with a structure of right bialgebroid. We introduce dynamical trace and use it for constructing central elements of $\tilde {\mathcal{K}}$.
Solutions to graded reflection equation of GL-type
本文列出了GL型量子超群基本向量表示对应的分次反射方程的所有解。
We list characters (one-dimensional representations) of the reflection equation algebra associated with the fundamental vector representation of the Drinfeld-Jimbo quantum group $\U_q\bigl(gl(n)\bigr)$.
We study relations between the two-parameter $\U_q(sl(n))$-invariant deformation quantization on $sl^*(n)$ and the reflection equation algebra. The latter is described by a quantum permutation on $\End(\C^n)$ given explicitly. The reflection equation algebra is used for constructing the one-parameter quantization on coadjoint orbits, including symmetric and certain bisymmetric and nilpotent ones. Our approach is based on embedding the quantized function algebras on the orbits into the algebra of functions on the quantum group $SL_q(n)$ via reflection equation algebra characters.
Quantum sphere is introduced as a quotient of the so-called Reflection Equation Algebra. This enables us to construct some line bundles on it by means of the Cayley-Hamilton identity whose a quantum version was discovered in \cite{PS}, \cite{GPS}. A new way to introduce some elements of "braided geometry" on the quantum sphere is discussed.
A solution to the reflection equation associated to a coideal subalgebra of $U_q(A_{2n-1}^{(1)})$ of type AII in the symmetric tensor representations is presented. If parameters of the coideal subalgebra are suitably chosen, the $K$ matrix does not depend on the quantum parameter $q$ and still agrees with a solution in arXiv:nlin/0411044 at $q=0$.
Dynamical Reflection Maps
本文运用范畴论构造了与动力杨-巴克斯特映射相关的动力反射映射（一类反射方程的解），并探讨了反射方程的箭袋论解，丰富了相关代数结构的研究。
By using the notion of a quantum double we introduce analogs of partial derivatives on a Reflection Equation algebra, associated with a Hecke symmetry of GL(N) type. We construct the matrix L=MD, where M is the generating matrix of the Reflection Equation algebra and D is the matrix composed of the quantum partial derivatives and prove that the matrices M, D and L satisfy a matrix identity, called the matrix Capelli one. Upon applying the quantum trace, it becomes a scalar relation, which is a far-reaching generalization of the classical Capelli identity. Also, we get a generalization of the s…
We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that $\La_R$ is a module algebra over the twisted tensor square \twist{$\Ha$}{$\Ha$} and the double $\D(\Ha)$. We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.
We discuss the consistency of the axioms which the definition of quantum Lie algebras is usually based on.
Reflections to set-theoretic solutions of the Yang-Baxter equation
本文通过探索双射非退化解与导出解的联系，确定杨-巴克斯特方程的反射，简化了特定情况下的反射研究，扩展了对合非退化解的已有结果至更一般的解，并提供了定义反射的方法及小阶斜括相关的数值数据。
Generalized Harish-Chandra morphism on Reflection Equation algebras
本文将广义 Harish-Chandra 态射从包络代数 U(gl(N)) 推广到反射方程代数，通过引入与斜可逆 Hecke 对称对应的生成矩阵特征值，并利用该态射构造权系统的量子类比。
Quantum Schur-Weyl duality and q-Frobenius formula related to Reflection Equation algebras
本文建立以 Hecke 代数和 Reflection Equation 代数为核心的量子 Schur-Weyl 对偶性，并构造 Schur 多项式与幂和的类似物，证明经由 Hecke 代数特征标的 q-Frobenius 公式。
Let R: V x V -> V x V be a Hecke type solution of the quantum Yang-Baxter equation (a Hecke symmetry). Then, the Hilbert-Poincre' series of the associated R-exterior algebra of the space V is a ratio of two polynomials of degree m (numerator) and n (denominator). Assuming R to be skew-invertible, we define a rigid quasitensor category SW(V) of vector spaces, generated by the space V and its dual V*, and compute certain numerical characteristics of its objects. Besides, we introduce a braided bialgebra structure in the modified Reflection Equation Algebra, associated with R, and equip objects o…
The reflection equation of Cherednik is a counterpart to the celebrated Yang-Baxter equation, with importance in the theory of integrable systems. We obtain several new solutions of the reflection equation using braces building on the work of Smoktunowicz, Vendramin and Weston. In particular, we show that every brace yields several simple solutions and that a natural class of solutions is a near-ring. We find more solutions for factorizable rings. Some of our solutions apply to the original parameter-dependent equation.
Given a right-non-degenerate set-theoretic solution $(X,r)$ to the Yang-Baxter equation, we construct a whole family of YBE solutions $r^{(k)}$ on $X$ indexed by its reflections $k$ (i.e., solutions to the reflection equation for $r$). This family includes the original solution and the classical derived solution. All these solutions induce isomorphic actions of the braid group/monoid on $X^n$. The structure monoids of $r$ and $r^{(k)}$ are related by an explicit bijective $1$-cocycle-like map. We thus turn reflections into a tool for studying YBE solutions, rather than a side object of study. …
The correspondence between the braid group on a solid torus of arbitrary genus and the algebra of Yang-Baxter and reflection equation operators is shown. A representation of this braid group in terms of $R$-matrices is given. The characteristic equation of the reflection equation matrix is considered as an additional skein relation. This could lead to an intrinsic definition of invariant link polynomials on solid tori and, via Heegaard splitting, to invariant link polynomials on arbitrary three-manifolds without boundary.
We propose a construction of the spherical subalgebra of a symplectic reflection algebra of an arbitrary rank corresponding to a star-shaped affine Dynkin diagram. Namely, it is obtained from the universal enveloping algebra of a certain semi-simple Lie algebra by the process of quantum Hamiltonian reduction. As an application, we propose a construction of finite-dimensional representations of the spherical subalgebra.
We introduce a representation theory of q-Lie algebras defined earlier in \cite{DG1}, \cite{DG2}, formulated in terms of braided modules. We also discuss other ways to define Lie algebra-like objects related to quantum groups, in particular, those based on the so-called reflection equations. We also investigate the truncated tensor product of braided modules.
Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov-Chekhov-Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are extendible to the non-constant case. In order to simplify future constructions of spin-chain Hamiltonians, a parametrization procedure is applied explicitly to all elements of the semi-dynamical reflectio…
We express the defining relations of the $q$-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect the quantum Lorentz group action in a straightforward way.
On a Small Version of the Reflection Equation Algebra
本文给出了与GL_N在奇偶单位根处相关的小反射方程代数的另一种表示，并利用该表示分类了其不可约表示，随后推广到一族代数并应用于U_q(sl_N)融合范畴的模范畴研究。
The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in terms of direct products of quantum algebra generators are derived, they lead to a representation of this braid group in terms of $R$-matrices. Restriction to the reflection equation operators only gives the coloured braid group. The reflection equation operators, describing the effect of handles attached to a 3-ball, satisfy characteristic equations which g…
We discuss how properties of Hecke symmetry (i.e., Hecke type R-matrix) influence the algebraic structure of the corresponding Reflection Equation (RE) algebra. Analogues of the Newton relations and Cayley-Hamilton theorem for the matrix of generators of the RE algebra related to a finite rank even Hecke symmetry are derived.
Let $R$ be a Hecke solution to the Yang-Baxter equation and $K$ be a reflection equation matrix with coefficients in an associative algebra $\A$. Let $R(x)$ be the baxterization of $R$ and suppose that $K$ satisfies a polynomial equation with coefficients in the center of $\A$. We construct solutions to the reflection equation with spectral parameter relative to $R(x)$, in the form of polynomials in $K$.
The set-theoretical reflection equation and its solutions, the reflection maps, recently introduced by two of the authors, is presented in general and then applied in the context of quadrirational Yang-Baxter maps. We provide a method for constructing reflection maps and we obtain a classification of solutions associated to all the families of quadrirational Yang-Baxter maps that have been classified recently.
We consider the reflection equation algebra for a finite dimensional R-matrix for the $(h,w)$-deformed Heisenberg algebra ${\cal U}_{h,w}(h(4))$. A representation of the reflection matrix $K$ is constructed using the matrix generators $L^{(\pm)}$ of the ${\cal U}_{h,w}(h(4))$ algebra. A series of representations of the K-matrix then may be generated by using the coproduct rules of the ${\cal U}_{h,w}(h(4))$ algebra. The complementary condition necessary for combining two distinct solutions of the reflection equation algebra yields the braiding relations between these two sets of generators. Th…
A skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang-Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.
Representation theory of the reflection equation algebra III: Classification of irreducible representations
本文发展最高权理论，分类反射方程代数（由 $GL(N,\mathbb{C})$ 的 $q$-变形 $R$-矩阵构造）的不可约有界 $*$-表示。
We use ring-theoretic methods and methods from the theory of skew braces to produce set-theoretic solutions to the reflection equation. We also use set-theoretic solutions to construct solutions to the parameter-dependent reflection equation.
We propose a classification of the solutions K to the semi-dynamical reflection equation with constant rational structure matrices associated to rational scalar Ruijsenaars-Schneider model. Four sets of solutions are identified and simple analytic transformations generate all solutions from these sets.
We classify non-operatorial matrices K solving Skylanin's quantum reflection equation for all R-matrices obtained from the newly defined general rank- n Hadamard type representations of the Temperley-Lieb algebra $TL_N(\sqrt n)$. They are characterized by a universal set of algebraic equations in a specific canonical basis uniquely defined from the "Master matrix" associated to the chosen realization of Temperley-Lieb algebra
In this article the quantized matrix algebras as in the title have been studied at a root of unity. A full classification of simple modules over such quantized matrix algebras of rank $2$ along with a class of finite dimensional indecomposable modules are given.
We consider intertwining relations of the triangular $q$-Onsager algebra, and obtain generic triangular boundary $K$-operators in terms of the Borel subalgebras of $U_{q}(sl_2)$. These $K$-operators solve the reflection equation.
We formulate a quantized reflection equation in which $q$-boson valued $L$ and $K$ matrices satisfy the reflection equation up to conjugation by a solution to the Isaev-Kulish 3D reflection equation. By forming its $n$-concatenation along the $q$-boson Fock space followed by suitable reductions, we construct families of solutions to the reflection equation in a matrix product form connected to the 3D integrability. They involve the quantum $R$ matrices of the antisymmetric tensor representations of $U_p(A^{(1)}_{n-1})$ and the spin representations of $U_p(B^{(1)}_{n})$, $U_p(D^{(1)}_{n})$ and …
The theory of generalized Weyl algebras is used to study the $2\times 2$ reflection equation algebra $\mathcal{A}=\mathcal{A}_q(\operatorname{M}_2)$ in the case that $q$ is not a root of unity, where the $R$-matrix used to define $\mathcal{A}$ is the standard one of type $A$. Simple finite dimensional $\mathcal{A}$-modules are classified, finite dimensional weight modules are shown to be semisimple, $\operatorname{Aut}(\mathcal{A})$ is computed, and the prime spectrum of $\mathcal{A}$ is computed along with its Zariski topology. Finally, it is shown that $\mathcal{A}$ satisfies the Dixmier-Moe…
Reflection equation for the scattering of lines moving in half-plane is obtained. The corresponding geometric picture is related with configurations of half-planes touching the boundary plane in 2+1 dimensions. This equation can be obtained as an additional to the tetrahedron equation consistency condition for a modified Zamolodchikov algebra.
Quadratic algebras related to the reflection equations are introduced. They are quantum group comodule algebras. The quantum group $F_q(GL(2))$ is taken as the example. The properties of the algebras (center, representations, realizations, real forms, fusion procedure etc) as well as the generalizations are discussed.
We explore the reflection-transmission quantum Yang-Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow to establish a direct relationship with the different previous works on the subject and make evident the advantages of the reflection-transmission algebra as an universal approach to integrable systems with impurities.
Non-polynomial Baxterized solutions of reflection equations associated with affine Hecke and affine Birman-Murakami-Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed.
A trick to obtain a systematic solution to the set-theoretical reflection equation is presented from a known one to the Yang-Baxter equation. Examples are given from crystals and geometric crystals associated to the quantum affine algebra of type $A^{(1)}_{n-1}$.
We study solutions of the reflection equation associated with the quantum affine algebra $U_{q}(\hat{gl}(N))$ and obtain diagonal K-operators in terms of the Cartan elements of a quotient of $U_{q}(gl(N))$. We also consider intertwining relations for these K-operators and find an augmented q-Onsager algebra like symmetry behind them.
We describe the diagonal reduction algebra D(gl(n)) of the Lie algebra gl(n) in the R-matrix formalism. As a byproduct we present two families of central elements and the braided bialgebra structure of D(gl(n)).
We present a new solution to the reflection equation associated with a coideal subalgebra of $U_q(A^{(1)}_{n-1})$ in the symmetric tensor representations and their dual. Elements of the $K$ matrix are expressed by a matrix product formula involving terminating $q$-hypergeometric series in $q$-boson generators. At $q=0$, our result reproduces a known set-theoretical solution to the reflection equation connected to the crystal base theory.
We introduce a family of polynomials in $q^2$ and four variables associated with the quantized algebra of functions $A_q(C_2)$. A new formula is presented for the recent solution of the 3D reflection equation in terms of these polynomials specialized to the eigenvalues of the $q$-oscillator operators.
A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of the parafermionic nature. We demonstrate that finite-dimensional representations are representations of deformed parafermionic algebra with internal Z_2-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supply us with irreducible representation of osp(1|2) superalgebra. We show that the…
We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$. We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations for the partition function.
Presentations for small reflection equation algebras of type A
本研究通过生成元和关系，给出A型GLₙ和SLₙ反射方程小代数的表现形式，将其表为无限维对应代数的商，且该形式对代数的适当整形式有效。
We show that the Painlev{é} VI equation has an equivalent form of the non-autonomous Zhukovsky-Volterra gyrostat. This system is a generalization of the Euler top in $C^3$ and include the additional constant gyrostat momentum. The quantization of its autonomous version is achieved by the reflection equation. The corresponding quadratic algebra generalizes the Sklyanin algebra. As by product we define integrable XYZ spin chain on a finite lattice with new boundary conditions.
We establish an explicit algebra isomorphism between the quantum reflection algebra for the $U_q(\hat{sl_2})$ R-matrix and a new type of current algebra. These two algebras are shown to be two realizations of a special case of tridiagonal algebras (q-Onsager).
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。
Solutions of 3D Reflection Equation from Quantum Cluster Algebra Associated with Symmetric Butterfly Quiver
本文基于对称蝴蝶有向图相关的量子簇代数，提出了一种新的3D反射方程解方法，通过引入多个量子二重对数来构造R-算子和K-算子。
Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The generators of the Askey-Wilson algebra are implemented to construct an operator valued $K$- matrix, a solution of a spectral dependent reflection equation. We consider the open driven diffusive system where the Askey-Wilson algebra arises as a boundary symmetry and can be used for an exact solution of t…
In this paper, we study transition matrices of PBW bases of the nilpotent subalgebra of quantum superalgebras associated with all possible Dynkin diagrams of type A and B in the case of rank 2 and 3, and examine relationships with three-dimensional (3D) integrability. We obtain new solutions to the Zamolodchikov tetrahedron equation via type A and the 3D reflection equation via type B, where the latter equation was proposed by Isaev and Kulish as a 3D analog of the reflection equation of Cherednik. As a by-product of our approach, the Bazhanov-Sergeev solution to the Zamolodchikov tetrahedron …
Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schrödinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed.…
Triangular solutions to the reflection equation for $U_q(\widehat{sl_n})$
本文研究 $U_q(\widehat{sl_n})$ 的反射方程，构造固定边界随机可积顶点模型及任意自旋对称张量表示的上下三角解，并证明 Ising 型模型的星-星关系。
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