# Reflection $K$-matrices for a nineteen vertex model with $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]$ symmetry

## TL;DR

This paper finds three classes of solutions to the boundary Yang-Baxter equation for a supersymmetric nineteen vertex model based on a twisted quantum affine Lie superalgebra, detailing their structures and free parameters.

## Contribution

It introduces three novel classes of reflection K-matrices for the nineteen vertex model with $U_q[osp(2|2)^{(2)}]$ symmetry, expanding understanding of boundary conditions in such models.

## Key findings

- Three classes of solutions to the boundary Yang-Baxter equation identified.
- Type I solution has three boundary free-parameters with all non-zero elements.
- Type II solution is even with one boundary free-parameter and null odd elements.

## Abstract

We derive the solutions of the boundary Yang-Baxter equation associated with a supersymmetric nineteen vertex model constructed from the three-dimensional representation of the twisted quantum affine Lie superalgebra $U_{q}[\mathrm{osp}\left(2|2\right)^{\left(2\right)}]\simeq U_{q}[C\left(2\right)^{\left(2\right)}]$. We found three classes of solutions. The type I solution is characterized by three boundary free-parameters and all elements of the corresponding reflection $K$-matrix are different from zero. In the type II solution, the reflection $K$-matrix is even (every element of the $K$-matrix with an odd parity is null) and it has only one boundary free-parameter. Finally, the type III solution corresponds to a diagonal reflection $K$-matrix with two boundary free-parameters.