  Starting from the classical r-matrix of the non-standard (or Jordanian)
quantum deformation of the sl(2,R) algebra, new triangular quantum deformations
for the real Lie algebras so(2,2), so(3,1) and iso(2,1) are simultaneously
constructed by using a graded contraction scheme; these are realized as
deformations of conformal algebras of (1+1)-dimensional spacetimes. Time-type
and space-type quantum algebras are considered according to the generator that
remains primitive after deformation: either the time or the space translation,
respectively. Furthermore by introducing differential-difference conformal
realizations, these families of quantum algebras are shown to be the symmetry
algebras of either a time or a space discretization of (1+1)-dimensional (wave
and Laplace) equations on uniform lattices; the relationship with the known Lie
symmetry approach to these discrete equations is established by means of twist
maps.
