Integrability of open spin chains with quantum algebra symmetry

In this paper, the authors prove that certain quantum spin chains with quantum-algebra symmetry are integrable. Specifically these are the quantum-algebra-invariant open chains associated with the affine Lie algebras A1(1), A2n(2), A2n-1(2), Bn(1), Cn(1), and Dn(1) in the fundamental representation. Conspicuously absent from this list is An(1) for n > 1. This is because in order to demonstrate integrability, the authors assume that the corresponding R matrix has crossing symmetry, which is not true in the case An(1) for n > 1