Solutions of the reflection equation for face and vertex models associated with $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$

TL;DR

This paper finds new diagonal solutions to the reflection equation for elliptic models related to affine Lie algebras, unifies known solutions, and explores their relations and limits to vertex models.

Contribution

It provides a comprehensive set of diagonal solutions for models associated with multiple affine Lie algebras, including new solutions in the trigonometric limit.

Findings

Recovered all known diagonal solutions for the considered algebras.

Established relations among solutions in the elliptic regime.

Derived new solutions for vertex models in the trigonometric limit.

Abstract

We present new diagonal solutions of the reflection equation for elliptic solutions of the star-triangle relation. The models considered are related to the affine Lie algebras $A_n^{(1)},B_n^{(1)},C_n^{(1)},D_n^{(1)}$ and $A_n^{(2)}$. We recover all known diagonal solutions associated with these algebras and find how these solutions are related in the elliptic regime. Furthermore, new solutions of the reflection equation follow for the associated vertex models in the trigonometric limit.