On the classification of rational K-matrices

Abstract

This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the classical limit (the spectral parameter goes to infinity). This derivation uses only the boundary Yang-Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are $\\mathfrak{g}$ and $\\mathfrak{h}$ then there exists a Lie-algebra involution of $\\mathfrak{g}$ for which the invariant sub-algebra is $\\mathfrak{h}$. In addition, we study some K-matrices which are not invertible in the classical limit . It is shown that their symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras.