Jon Links
Consider the statement "Every Yang-Baxter integrable system is defined to be exactly-solvable". To formalise this statement, definitions and axioms are introduced. Then, using a specific Yang-Baxter integrable bosonic system, it is shown that a paradox emerges. A generalisation for completely integrable bosonic systems is also developed.
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Scalene Yang-Baxter maps and Lax triples
本文研究了Yang-Baxter方程的推广，并探讨了其解与矩阵重构问题的联系，提出斜方Yang-Baxter映射的概念，并构造了与KdV和NLS类型可积方程相关的映射。
Starting from known solutions of the functional Yang-Baxter equations, we exhibit Miura type of transformations leading to various known integrable quad equations. We then construct, from the same list of Yang-Baxter maps, a series of non-autonomous solvable recurrences of order two.
We prove that one-dimensional elastic relativistic collisions satisfy the set-theoretical Yang-Baxter equation. The corresponding collision maps are symplectic and admit a Lax representation. Furthermore, they can be considered as reductions of a higher dimensional integrable Yang-Baxter map on an invariant manifold. In this framework, we study the integrability of transfer maps that represent particular periodic sequences of collisions.
We recently introduced a class of ${\mathbb{Z}}_N$ graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper we introduce the corresponding Yang-Baxter maps. Many well known examples belong to this scheme for $N=2$, so, for $N\geq 3$, our systems may be regarded as generalisations of these. In particular, for each $N$ we introduce a generalisation of the map $H_{III}^B$ in the classification of scalar Yang-Baxter maps. For $N=3$ this is equivalent to the Yang-Baxter map associated with the discrete modified Boussinesq equation. For $N\g…
In this paper we show that there are explicit Yang-Baxter maps with Darboux-Lax representation between Grassmann extensions of algebraic varieties. Motivated by some recent results on noncommutative extensions of Darboux transformations, we first derive a Darboux matrix associated with the Grassmann-extended derivative Nonlinear Schrodinger (DNLS) equation, and then we deduce novel endomorphisms of Grassmann varieties, which possess the Yang-Baxter property. In particular, we present ten-dimensional maps which can be restricted to eight-dimensional Yang-Baxter maps on invariant leaves, related…
Dispersionless modified DKP hierarchy as the Yang-Baxter equation
研究修正DKP层次无弥散版本，通过将其定义为广田-三和田型τ函数关系极限，重新表述为8顶点模型玻尔兹曼权重巴克斯特R矩阵的杨-巴克斯特方程。
We introduce four lists of families of non-abelian quadrirational Yang-Baxter maps.
On refactorization problems and rational Lax matrices of quadrirational Yang-Baxter maps
本文为四有理Yang-Baxter映射构造有理Lax表示，利用对称性导出多类映射的重分解问题，并给出非对合多参数示例。
We present Yang-Baxter maps associated to elliptic curves. They are related to discrete versions of the Krichever-Novikov and the Landau-Lifshits equations. A lifting of scalar integrable quad-graph equations to two-field equations is also shown.
Two-component Yang-Baxter maps and star-triangle relations
本文提出从经典星-三角关系和量子Yang-Baxter方程直接推导双分量Yang-Baxter映射，发现其满足单一映射和交织映射两种方程，并由此推导出十六个不同映射。
According to Shibukawa, ternary systems defined on quasigroups and satisfying certain conditions provide a way of constructing dynamical Yang-Baxter maps. After noticing that these conditions can be interpreted as 3-dimensional compatibility of equations on quad-graphs, we investigate when the associated dynamical Yang-Baxter maps are in fact parametric Yang-Baxter maps. In some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints. Conversely, parametric YB maps on quasigroups with an invariance condition give rise to 3-dimensional compatible…
Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter equation') are considered. A birational map $(x,y)\mapsto(u,v)$ is called quadrirational, if its graph is also a graph of a birational map $(x,v)\mapsto(u,y)$. We obtain a classification of quadrirational maps on $\CP^1\times\CP^1$, and show that all of them satisfy the Yang-Baxter equation. These maps possess a nice geometric interpretation in terms of linear pencil of conics, the Yang-Baxter property being interpreted as a new incidence theorem of the projective geometry of conics.
Scalene Yang--Baxter triples as a source of hidden symmetries beyond the ordinary Yang--Baxter equation
该研究以不等边Yang-Baxter三元组为基础构造非厄米自旋链，发现其转移矩阵存在交叉对易性，揭示了普通Yang-Baxter方程框架外的隐藏对称性来源。
For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression o…
We propose a new generalization of the Yang-Baxter equation, where the R-matrix depends on cluster $y$-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the recently-found correspondence between Yang-Baxter equations and supersymmetric gauge theories. The $S^2$ partition function of a certain 2d $\mathcal{N}=(2,2)$ quiver gauge theory gives an R-matrix, whereas its FI parameters can be identified with the cluster $y$-variables.
We introduce an odd supersymmetric version of the Kronecker elliptic function. It satisfies the genus one Fay identity and supersymmetric version of the heat equation. As an application we construct an odd supersymmetric extensions of the elliptic $R$-matrices, which satisfy the classical and the associative Yang-Baxter equations.
All 4 x 4 solutions of the quantum Yang-Baxter equation
本文完成对4×4Yang-Baxter方程解的分类，发现非规则解并研究其与量子Yang-Baxter方程的关系，揭示规则解与非规则解之间的对应关系差异。
We give a direct proof of the fact that elliptic solutions of the associative Yang-Baxter equation arise from appropriate spherical orders on an elliptic curve.
We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.
We find all non-equivalent constant solutions for classical associative Yang-Baxter equation for $gl(3)$. New examples found in the classification yield the corresponding quadratic trace Poisson brackets, double Poisson structures on free associative algebra with three generators and anti-Frobenius associative algebras.
A new method for solving the Yang-Baxter equation is presented. It is based on the Taylor expansion of R-matrix which is developed up to the power lambda^6. Using this method the R-matrix for integrable spin ladder is calculated.
Some New Solutions of Yang-Baxter Equation
将杨-巴克斯特方程改写为矩阵向量空间中的三重积方程，得到了有理型和三角型的新解。
We consider a matrix refactorization problem, i.e., a "Lax representation", for the Yang-Baxter map that originated as the map of polarizations from the "pure" 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang-Baxter equation, but satisfies a mixed version of the Yang-Baxter equation together with the Yang-Baxter map. Such maps have been called "entwining Yang-Baxter maps" in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a mat…
Reductions for systems of ODEs integrable via the standard factorization method (the Adler-Kostant-Symes scheme) or the generalized factorization method, developed by the authors earlier, are considered. Relationships between such reductions, operator Yang-Baxter equations, and some kinds of non-associative algebras are established.
In this work we consider a functional method in the theory of exactly solvable models based on the Yang-Baxter algebra. Using this method we derive the eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions for general values of the anisotropy and boundary parameters.
Yang--Baxter maps of KdV, NLS and DNLS type on division rings
本研究构造了除环上与KdV、NLS及导数NLS方程相关的非交换Yang--Baxter方程集合论解，证明部分映射的Liouville可积性与约化性质，并通过非交换Darboux矩阵重分解构造验证解。
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It is shown that the Yang-Baxter variables can be chosen as invariants of the multi-parameter symmetry groups of the equations. We use the classification results by Adler, Bobenko and Suris to demonstrate this method. Some new examples of Yang-Baxter maps are derived in this way from multi-field integrable equations.
It is well known that, given a Yang-Baxter map, there is a hierarchy of commuting transfer maps, which arise out of the consideration of initial value problems. In this paper, we show that one can construct invariants of the transfer maps corresponding to the $n$-periodic initial value problem on the two-dimensional lattice, using the same generating function that is used to produce invariants of the Yang-Baxter map itself. Moreover, we discuss the Liouville integrability of these transfer maps. Finally, we consider four-dimensional Yang-Baxter maps corresponding to the nonlinear Schrödinger (…
The duality of dual resonance models is shown to imply that the four point string correlation function solves the Yang-Baxter equation. A reduction of transfer matrices to $A_l$ symmetry is described by a restriction of the KP $τ$ function to Toda molecules.
On the growth of operator entanglement in brickwork circuits with Yang--Baxter gates
该研究针对带杨-巴克斯特门的一维砖墙电路，建立了算子施密特秩的多种增长上界，构造了反例证明部分情形下的指数增长，相关结果为算子纠缠增长问题提供了重要进展。
Motivated by the proof of Rump of a conjecture of Gateva-Ivanova on the decomposability of square-free solutions to the Yang-Baxter equation, we present several other decomposability theorems based on the cycle structure of a certain permutation associated with the solution.
On the integrability of N=2 Landau-Ginzburg models: A graph generalization of the Yang-Baxter equation
对N=2的Landau-Ginzburg模型的可积性的研究自然地导致了Yang-Baxter方程的图形推广，它综合了著名的顶点和RSOS Yang-Baxter方程。对于A模型的$t_2$微扰，我们找到了这个方程的一个非平凡解，它与手征Potts模型的玻尔兹曼权密切相关.
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