Reflection matrices with $U_q[os{p}^{(2)}(2|2m)]$ symmetry

Abstract

We propose a classification of the reflection $K$-matrices (solutions of the boundary Yang-Baxter equation) for the $U_q[{\mathrm{osp}}^{\left(2\right)}\left(2|2m\right)]=U_q[C^{\left(2\right)}\left(m+1\right)]$ vertex-model. We have found four families of solutions, namely, the complete solutions, in which no elements of the reflection $K$-matrix is null, the block-diagonal solutions, the $X$-shape solutions and the diagonal solutions. We highlight that these diagonal $K$-matrices also hold for the $U_q[{\mathrm{osp}}^{\left(2\right)}\left(2n+2|2m\right)]=U_q[D^{\left(2\right)}\left(n+1,m\right)]$ vertex-model.

24 pages. Abstract and introduction rewriten, references added, results unchanged. Keywords: Integrable models, boundary Yang-Baxter equation, reflection $K$-matrices, twisted Lie superalgebras, orthosymplectic algebras