Related papers: Reflection-Transmission Quantum Yang-Baxter Equati…
The definitions of the main notions related to the quantum inverse scattering methods are given. The Yang-Baxter equation and reflection equations are derived as consistency conditions for the factorizable scattering on the whole line and…
To the Yang-Baxter equation an additional relation can be added. This is the reflection equation which appears in various places, with or without spectral parameter. For example, in factorizable scattering on a half-line, integrable lattice…
Connections between set-theoretic Yang-Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic $R$-matrices are expressed as twists of known solutions. We then focus on reflection and…
We propose a new dynamical reflection algebra, distinct from the previous dynamical boundary algebra and semi-dynamical reflection algebra. The associated Yang-Baxter equations, coactions, fusions, and commuting traces are derived. Explicit…
Inspired by factorized scattering from delta-type impurities in (1+1)-dimensional space-time, we propose and analyse a generalization of the Zamolodchikov-Faddeev algebra. Distinguished elements of the new algebra, called reflection and…
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…
We consider the scattering problems of a quantum particle in a system with a single Y-junction and in ring systems with double Y-junctions. We provide new formalism for such quantum mechanical problems. Based on a path integral approach, we…
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…
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…
We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the…
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…
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the…
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…
We establish that the Yang-Baxter equations in the presence of an impurity can in general only admit solutions of simultaneous tranmission and reflection when the transmission and reflection amplitudes commute in the defect degrees of…
Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from four-dimensional gauge theory. In the present paper, we extend this picture, fill in…
The reflection equations (RE) are a consistent extension of the Yang-Baxter equations (YBE) with an addition of one element, the so-called reflection matrix or $K$-matrix. For example, they describe the conditions for factorizable…
Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schr\"odinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on…
We formulate a systematic construction of commuting quantum traces for reflection algebras. This is achieved by introducing two sets of generalized reflection equations with associated consistent fusion procedures. Products of their…
In this article we review the basic concepts regarding quantum integrability. Special emphasis is given on the algebraic content of integrable models. The associated algebras are essentially described by the Yang-Baxter and boundary…
The main aim of this paper is to determine reflections to bijective and non-degenerate solutions of the Yang-Baxter equation, by exploring their connections with their derived solutions. This is motivated by a recent description of left…