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

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.

6 pages. Keywords: reflection K-matrices, boundary Yang-Baxter equation, Lie superalgebras, twisted Lie algebras, nineteen vertex-models