SOLVING COLORED YANG-BAXTER EQUATION BY WU’S METHOD

In this article, we discuss nonsymmetric solutions of the colored Yang-Baxter equation dependent on spectral as well as colored parameters and give all seven-vertex solutions by Wu＇s method. It is also proved that the solutions are composed of six groups of basic solutions up to five solution transformations. Moreover, al l solutions can be classified into two categories called Baxter type and free-fermion type.

Rota-Baxter Operators on 3-dimensional Lie Algebras and Solutions of the Classical Yang-Baxter Equation

In this paper,we compute Rota-Baxter operators on the 3-dimensional Lie algebra g whose derived algebra’s dimension is 2.Furthermore,we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ■ _(ad~*) g~* and some new structures of left-symmetric algebra induced from g and its Rota-Baxter operators.

The Singular Upper Triangle Type Solutions with Spin 1/2 of Quantum Yang-Baxter Equation (I)

In these two papers (I) and (II), the singular upper triangle type solutions with spin 1/2 of quantum Yang-Baxter equation are given. In the Paper (I), we give the Yang-Baxter equation and give the general solutions of some function equations for the next paper.

QUANTUM COMODULE APPROACH THE SOLUTION OF THE QUANTUM YANG-BAXTER EQUATION

In this paper, the relationship between solutions of the Quantum Yang-Baxter Equation and quantum comodules, and some properties of the quantum comodule category are characterized here. These results make it possible to give some set-theoretical solutions of the Quantum Yang-Baxter Equation.

The Singular Upper Triangle Type Solutions with Spin 1/2 of Quantum Yang-Baxter Equation （Ⅱ）

In these two papers (Ⅰ) and (Ⅱ),the singular upper triangle type solutions with spin 1/2 of quantum Yang-Baxter equation are given.In the Paper (Ⅱ),we give the general solutions of the Yang-Baxter equation in this case.

On Set—Theoretical Solutions to Quantum Yang—Baxter Equation

The problem on the set-theoretical solutions to the quantum Yang-Baxter equation was presented byDrinfel'd as a main unsolved problem in quantum group theory. The set-theoretical solutions are a natural extensionof the usual (linear) solutions. In this paper, we not only give a further study on some known set-theoretical solutions(the Venkov's solutions), but also find a new kind of set-theoretical solutions which have a geometric interpretation.Moreover, the new solutions lead to the metahomomorphisms in group theory.

Solution to the Yang-Baxter Equation and QuantumYang-Baxter H-comodules

Let H be an arbitrary Hopf algebra over a field k. In this paper, at first wedeal with the relationship between solutions to the Yang-Baxter equation and quantumYang-Baxter H-comodules; then we use the results to give a solution to the Yang-Baxterequation over H.

Solutions of Yang-Baxter equation with color parameters

All solutions of Yang-Baxter equation with color parameters for six-vertex model are achieved by five solution transformations, three nondegenerate basic solutions and several degenerate basic solutions. It is shown that all solutions of Yang-Baxter equation with spectral parameter for six-vertex model are obtainable from solutions of Yang-Baxter equation with color parameters for six-vertex model.

THE DIFFERENTIAL REALIZATION OF NEW SOLUTIONS FOR YANG-BAXTER EQUATION

I. INTRODUCTION Yang-Baxter equation plays a crucial role in the nonlinear integrable system. Its standard solutions can be constructed in terms of the universal R-matrix and representations of the corresponding quantum universal enveloping algebra (or called quantum algebra for short),

Another solution of Yang-Baxter equation on set and "metahomomorphisms on groups

1 Another solution of Yang-Baxter equation on setIN ref [1], Drinfeld said that up to now there exist just two examples on the solution R of theYang-Baxter

Yang-Baxter Equations With Reflection Associated With the Scattering From Kaleidoscope

YBE associated with the scattering from kaleidoscopes of Sutherland has been found. The reflection behaviour at the Dynkin diagram and its physical meaning are explored.

QUANTUM DOUBLE CONSTRUCTION OF NEW SOLUTIONS TO THE QUANTUM YANG-BAXTER EQUATION

By establishing a new Hopf algebra and the corresponding quantum double,we construct a new universal.R-matrix. Under concrete representations, it produces a family of new solutions to the quantum Yang-Baxter equation without a parameter,in which the so-called non-standard solution associated with the Lie algebra sl(3)is included.

Quantum Discord in Two-Qubit System Constructed from the Yang-Baxter Equation

Quantum correlations among parts of a composite quantum system are a fundamental resource for several applications in quantum information. In general, quantum discord can measure quantum correlations. In that way,we investigate the quantum discord of the two-qubit system constructed from the Yang–Baxter Equation. The density matrix of this system is generated through the unitary Yang–Baxter matrix R. The analytical expression and numerical result of quantum discord and geometric measure of quantum discord are obtained for the Yang–Baxter system. These results show that quantum discord and geometric measure of quantum discord are only connect with the parameter θ,which is the important spectral parameter in Yang–Baxter equation.

Some Rank Formulas for the Yang-Baxter Matrix Equation AXA=XAX

Let A be an arbitrary square matrix,then equation AXA=XAX with unknown X is called Yang-Baxter matrix equation.It is difficult to find all the solutions of this equation for any given A.In this paper,the relations between the matrices A and B are established via solving the associated rank optimization problem,where B=AXA=XAX,and some analytical formulas are derived,which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.

Quasi-linear Cycle Sets and the Retraction Problem for Set-theoretic Solutions of the Quantum Yang-Baxter Equation

Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.

经典杨-Baxter方程和sl(p,q)Gaudin模型

本文提出了一类与李超代数sl(p,q)相关的可积Gaudin 模型,并给出了其Hamiltonian 的明显形式.

LIE SUPER-BIALGEBRA STRUCTURES ON GENERALIZED SUPER-VIRASORO ALGEBRAS

In this article, Lie super-bialgebra structures on generalized super-Virasoro algebras/： are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if Hi（ ） = 0.

Lie Bialgebra Structures on Generalized Heisenberg-Virasoro Algebra

In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It is proved that all Lie bialgebra structures on centerless generalized Heisenberg-Virasoro algebra L are coboundary triangular by proving that the first cohomology group H1 (L,V) =0.

A 9×9 matrix representation of Temperley-Lieb algebra and corresponding entanglement

We present the matrix U. The 9 × 9 U matrix is a representation of specialized Temperley-Lieb algebra. Based on which, a unitary R matrix is generated via the Yang-Baxterization approach. The 9 × 9 R matrix, a solution of the Yang-Baxter equation, is universal for quantum entanglement.

A Characterization of Metahomomorphisms on Abelian Groups

In this paper, a characterization of metahomomorphisms on Abelian groups is given.