Exact Results in Quantum Theory

In this talk the notions of 3D-compatible, Yang-Baxter and Braid maps will be introduced, together with their interrelations andtheir connections to integrable difference systems. Furthermore,derivations and extensions of these maps to skew-fields will bediscussed. The latter will result to the extension and generalizationof integrable difference systems to totally non-commutating variables.In that respect we obtain the non-Abelian lattice-modifiedGel’fand-Dikii hierarchy,together with the explicit form of anon-Abelian hierarchy that we refer to as the lattice-NQC Gel’fand-Dikii hierarchy.