New quantum conformal algebras and discrete symmetries

Two new quantum deformations of the Lie algebras so(4,2) and so(5,1) are constructed as quantum conformal algebras of the Minkowskian and Euclidean spaces. These deformations are called either ‘mass-like’ or ‘length-like’ according to the dimensional properties of the deformation parameter. Their Hopf structure, universal R matrix and differential-difference realization are obtained in a unified setting by considering a contraction parameter related to the speed of light, which ensures a well defined non-relativistic limit to new quantum conformal Galilean algebras. These quantum conformal algebras are shown to be symmetry algebras of either time or space discretizations of wave/Laplace equations on uniform lattices. These results lead to a proposal for time and space discrete Maxwell equations with quantum algebra symmetry.