Four-dimensional quantum affine algebras and space–time q-symmetries

A global model of the q deformation for the quasiorthogonal Lie algebras generating the groups of motions of the four-dimensional affine Cayley–Klein (CK) geometries is obtained starting from the three-dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions either of the space–time or space symmetry algebras [Poincaré (3+1), Galilei (3+1), 4-D Euclidean as well as others] appear in this context as particular cases. For some of these classical algebras several q deformations are directly obtained.