The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes

Abstract

We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $\kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $\kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $\kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar\'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $\kappa$-Newtonian and $\kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $\kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $\kappa$, the curvature parameter $\eta$ and the speed of light parameter $c$.