  Two non-standard quantum deformations of the (1+1) Schr\"odinger algebra are
identified with the symmetry algebras of either a space or time uniform lattice
discretization of the Schr\"odinger 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
Schr\"odinger 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.
