Integrable quantum mappings.

We discuss the canonical structure of a class of integrable quantum mappings, i.e. iterative canonical transformations that can be interpreted as a discrete dynamical system. As particular examples we consider quantum map-pings associated with the lattice analogues of the KdV and MKdV equations. These mappings possess a non-ultralocal quantum Yang-Baxter structure leading to the existence of commuting families of exact quantum invariants. We derive the associated quantum Miura transformations between these mappings and the corresponding quantum bi-Hamiltonian structure.