New time-type and space-type non-standard quantum algebras and discrete symmetries

Starting from the classical r matrix of the non-standard (or Jordanian) qua ntum deformation of the sl(2, R) algebra, new triangular quantum deformatio ns for the real Lie algebras so(2, 2), so(3, 1) and iso(2, 1) are simultane ously constructed by using a graded contraction scheme; these are realized as deformations of conformal algebras of (1 + 1)-dimensional spacetimes. Ti me- and space-type quantum algebras are considered according to the generat or that remains primitive after deformation: either the time or the space t ranslation, respectively. Furthermore, by introducing differential-differen ce 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 relat ionship with the known Lie symmetry approach to these discrete equations is established by means of twist maps.