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 Poincaré 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 ($θ_{μν}$-deformation) and second with Lie-algebraic structure of quantum space-time, introducing so-called $κ$-deformation. The third fundamental constant of Nature - fundamental mass $κ$ or length $λ$ - appears naturally in proposed quantum relativistic symmetry scheme. The deformed Minkowski space is described as the representation space (Hopf-module) of deformed Poincaré algebra. Some possible perspectives of quantum-deformed relativistic symmetries will be outlined.