Transmission matrices in $\mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}}$ & ${{\mathfrak{U}}_q}\left( {\mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}}} \right)$ quantum spin chains

Abstract

The $\mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}}$ and ${{\mathfrak{U}}_q}\left( {\mathfrak{g}{{\mathfrak{l}}_{\mathcal{N}}}} \right)$ quantum spin chains in the presence of integrable spin impurities are considered. Within the Bethe ansatz formulation, we derive the associated transmission amplitudes, and the corresponding transmission matrices — representations of the underlying quadratic algebra — that physically describe the interaction between the various particle-like excitations displayed by these models and the spin impurity.