# On the classification of rational K-matrices

## TL;DR

This paper classifies the residual symmetries of rational K-matrices using boundary Yang-Baxter equations, confirming previous assumptions and exploring non-invertible cases with non-reductive symmetry algebras.

## Contribution

It provides a derivation of residual symmetries of rational K-matrices, confirming the existence of involutive automorphisms relating original and residual symmetry algebras.

## Key findings

- Residual symmetries correspond to involutive automorphisms of the Lie algebra
- Invertible K-matrices have reductive symmetry algebras
- Non-invertible K-matrices have semi-direct sum symmetry algebras

## 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.