Related papers: Dynamical Yang-Baxter Maps with an Invariance Cond…
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…
We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance…
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…
Yang-Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 x 2…
We generalize the FRT construction for the quiver-theoretical quantum Yang-Baxter equation and obtain a left bialgebroid $\mathfrak{A}(w)$. There are some relations between the left bialgebroid $ \mathfrak{A}(w)$ and a left bialgebroid…
A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice…
Notions of an (H, X)-bialgebroid and of its dynamical representation are proposed. The dynamical representations of each (H, X)-bialgebroid form a tensor category. Every dynamical Yang-Baxter map R(lambda) satisfying suitable conditions, a…
The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and/or transfer-matrices. The general…
Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of…
We construct novel solutions to the set-theoretical entwining Yang-Baxter equation. These solutions are birational maps involving non-commutative dynamical variables which are elements of the Grassmann algebra of order $n$. The maps arise…
It is shown how Yang-Baxter maps may be directly obtained from classical counterparts of the star-triangle relations and quantum Yang-Baxter equations. This is based on reinterpreting the latter equation and its solutions which are given in…
A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The…
In this article we propose an algebraic system, which is an abelian group $(A,+)$ with a family of non-associative and non-(left)distributive multiplications $\{\cdot_{\lambda}\}_{\lambda\in H}$. We call this algebraic system dynamical…
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…
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…
In this paper, by making use of category theory, we construct dynamical reflection maps, solutions to a version of the reflection equation associated with suitable dynamical Yang-Baxter maps, set-theoretic solutions to the braid relation…
For any algebra two families of coloured Yang-Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang-Baxter equation. An open problem about a system of functional equations is stated. The matrix forms…
We construct birational maps that satisfy the parametric set-theoretical Yang-Baxter equation and its entwining generalisation. For this purpose, we employ Darboux transformations related to integrable Nonlinear Schr\"odinger type equations…
We apply the fusion procedure to a quantum Yang-Baxter algebra associated with time-discrete integrable systems, notably integrable quantum mappings. We present a general construction of higher-order quantum invariants for these systems. As…
The quantum Yang-Baxter equation admits generalisations to systems of Yang-Baxter type equations called Yang-Baxter systems. Starting from algebra structures, we propose new constructions of some constant as well as the spectral-parameter…