On the semi-dynamical reflection equation: solutions and structure matrices

TL;DR

This paper constructs explicit solutions and parametrizations for the semi-dynamical reflection equation with rational structure matrices, revealing new solutions, parametrizations, and expressions for R-matrices relevant to integrable models.

Contribution

It provides the first explicit solutions and parametrizations for the semi-dynamical reflection equation with rational structure matrices, including new R-matrix expressions.

Findings

Only two sets of constant solutions extend to the non-constant case.

Explicit parametrization of all elements of the semi-dynamical reflection equation.

Discovery of new expressions for R-matrices and structure matrices.

Abstract

Explicit solutions of the non-constant semi-dynamical reflection equation are constructed, together with suitable parametrizations of their structure matrices. Considering the semi-dynamical reflection equation with rational non-constant Arutyunov-Chekhov-Frolov structure matrices, and a specific meromorphic ansatz, it is found that only two sets of the previously found constant solutions are extendible to the non-constant case. In order to simplify future constructions of spin-chain Hamiltonians, a parametrization procedure is applied explicitly to all elements of the semi-dynamical reflection equation available. Interesting expressions for `twists' and R-matrices entering the parametrization procedure are found. In particular, some expressions for the R-matrices seem to appear here for the first time. In addition, a new set of consistent structure matrices for the semi-dynamical reflection equation is obtained.