Deformed Quantum Relativistic Phase Spaces -- an Overview

Abstract

We describe three ways of modifying the relativistic Heisenberg algebra - first one not linked with quantum symmetries, second and third related with the formalism of quantum groups. The third way is based on the identification of generalized deformed phase space with the semidirect product of two dual Hopf algebras describing quantum group of motions and the corresponding quantum Lie algebra. As an example the $\kappa$-deformation of relativistic Heisenberg algebra is given, determined by $\kappa$-deformed D=4 Poincar\'{e} symmetries.