ABSTRACT
This chapter discusses a new type of vertex model will be constructed from the ancestor Lax operator using non-fundamental representations. It examines how these Lax operators can be generated in a systematic way confirming their integrability. The chapter focuses on the ultralocal systems due to their relative simplicity and examines a unifying scheme for generating quantum and statistical integrable models. It describes a new direction of study would be to investigate whether there could be any meaningful statistical model corresponding to the integrable non-ultralocal models. To describe an integrable system with such an involved structure, naturally one can no longer start from the Hamiltonian of the model as is customary in physics, since the Hamiltonian is merely one among many commuting conserved charges. Non-ultralocal models are mostly non-fundamental systems with the infinite dimensional representations defined in some Hilbert space.
