Integrable defects in affine Toda field theory and infinite dimensional representations of quantum groups

Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang-Baxter equations, and second by solving a linear intertwining relation between a finite dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite dimensional representation expressed in terms of sets of generalized `quantum' annihilation and creation operators. The principal examples analysed are based on the $a_2^{(2)}$ and $a_n^{(1)}$ affine Toda models but examples of similar infinite dimensional representations for quantum Borel algebras for all other affine Toda theories are also provided.