Constant Solutions of Reflection Equations and Quantum Groups

TL;DR

This paper investigates constant solutions to reflection equations associated with quantum groups, providing explicit solutions and quadratic algebra structures, extending the understanding of boundary conditions in integrable models.

Contribution

It introduces a classification of constant solutions to reflection equations that are compatible with quantum group coactions, including explicit algebraic structures.

Findings

Explicit constant solutions for known Yang-Baxter solutions

Quadratic algebras defined by reflection equations

Framework for boundary conditions in integrable models

Abstract

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 models with non-periodic boundary conditions, non-commutative differential geometry on quantum groups, etc. We study two forms of spectral parameter independent reflection equations, chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations. For a variety of known solutions of the Yang-Baxter equation we give the constant solutions of the reflection equations. Various quadratic algebras defined by the reflection equations are also given explicitly.