Quantum Deformations of Einstein's Relativistic Symmetries

Abstract

We shall outline two ways of introducing the modification of Einstein's relativistic symmetries of special relativity theory - the Poincar\'{e} symmetries. The most complete way of introducing the modifications is via the noncocommutative Hopf-algebraic structure describing quantum symmetries. Two types of quantum relativistic symmetries are described, one with constant commutator of quantum Minkowski space coordinates ($\theta_{\mu\nu}$-deformation) and second with Lie-algebraic structure of quantum space-time, introducing so-called $\kappa$-deformation. The third fundamental constant of Nature - fundamental mass $\kappa$ or length $\lambda$ - appears naturally in proposed quantum relativistic symmetry scheme. The deformed Minkowski space is described as the representation space (Hopf-module) of deformed Poincar\'{e} algebra. Some possible perspectives of quantum-deformed relativistic symmetries will be outlined.