Constant solutions of reflection equations and quantum groups

Abstract

To the Yang–Baxter equation an additional relation can be added. This is the reflection equation that appears in various places, with or without spectral parameter, e.g., in factorizable scattering on a half-line, integrable lattice models with nonperiodic boundary conditions, noncommutative differential geometry on quantum groups, etc. Two forms of spectral-parameter-independent reflection equations are studied, 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 the constant solutions of the reflection equations are given. Various quadratic algebras defined by the reflection equations are also given explicitly.