On quantum algebra symmetries of discrete Schrödinger equations

Abstract

Two non-standard quantum deformations of the (1+1) Schrödinger algebra are identified with the symmetry algebras of either a space or time uniform lattice discretization of the Schrödinger equation. For both cases, the deformation parameter of the corresponding Hopf algebra can be interpreted as the step of the lattice. In this context, the introduction of nonlinear maps defining Schrödinger and $sl(2,\R)$ quantum algebras with classical commutation rules turns out to be relevant. The problem of finding a quantum algebra linked to the full space-time discretization is also discussed.