Related papers: Dynamical Yang-Baxter Maps with an Invariance Cond…

A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a…

This paper deals with left non-degenerate set-theoretic solutions to the Yang-Baxter equation (=LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution is associated a shelf…

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the…

Based on the method which is given in Ref. [Sun et.al. arXiv:0904.0092v1], we present another $9\times 9$ unitary $\breve{R}-$matrix, solution of the Yang-Baxter Equation, is obtained in this paper. The entanglement properties of…

The aim of this paper is first to introduce and study Rota-Baxter cosystems and bisystems as generalization of Rota-Baxter coalgebras and bialgebras, respectively, with various examples. The second purpose is to provide an alternative…

We study ${\rm GL}_N$ rational $R$-matrix, which turns into the 11-vertex $R$-matrix in the $N=2$ case. First, we describe its relations to dynamical and semi-dynamical $R$-matrices using the IRF-Vertex type transformations. As a by-product…

We discuss homogeneous Yang-Baxter deformations of integrable sigma models in terms of twist operators. We show that the twist operators behave as the classical analogue of a Drinfeld twist, for all abelian and almost abelian deformations.…

A new class of indecomposable, irretractable, involutive, non-degenerate set-theoretic solutions of the Yang--Baxter equation is constructed. This class complements the class of such solutions constructed in \cite{CO22} and together they…

We consider the quiver Yangians associated to general affine Dynkin diagrams. Although the quivers are generically not toric, the algebras have some similar structures. The odd reflections of the affine Dynkin diagrams should correspond to…

In this article, a system of Yang-Baxter-type matrix equations is studied, $XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix Yang-Baxter equation and exhibits a broken symmetry. We investigate the solutions of this system from various…

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group…

Yang-Baxter bialgebras, as previously introduced by the authors, are shown to arise from a double crossproduct construction applied to the bialgebra R T T = T T R, E T = T E R, \Delta(T) = T \hat{\otimes} T, \Delta(E) = E \hat{\otimes} T +…

We classify rational solutions of a specific type of the set theoretical version of the pentagon equation. That is, we find all quadrirational maps $R:(x,y)\mapsto (u(x,y),v(x,y)),$ where $u, v$ are two rational functions on two arguments,…

We construct the lens hyperbolic modular double, a new algebraic structure whose intertwining operator produces a lens hyperbolic hypergeometric solution of the Yang--Baxter equation.

A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the…

For a given $(X,S,\beta)$, where $S,\beta\colon X\times X\to X\times X$ are set theoretical solutions of Yang-Baxter equation with a compatibility condition, we define an invariant for virtual (or classical) knots/links using non…

In this paper we propose versions of the associative Yang-Baxter equation and higher order $R$-matrix identities which can be applied to quantum dynamical $R$-matrices. As is known quantum non-dynamical $R$-matrices of Baxter-Belavin type…

We develop Yang-Baxter integrability structures connected with the quantum affine superalgebra Uq(\hat sl(2|1)). Baxter's Q-operators are explicitly constructed as supertraces of certain monodromy matrices associated with (q-deformed)…

This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields $F_{2^n}$. We define $R$-matrices using a map $f$ on $F_{2^n}$ and establish necessary and sufficient conditions for $f$ to satisfy the…

We define integrability preserving Yang-Baxter deformations of symmetric space sigma models with non-semi-simple symmetry group, in particular the flat space string, using only the essential structures of a symmetric space sigma model. For…