The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes

Abstract

We derive the non-relativistic c→∞ and ultra-relativistic c→0 limits of the κ-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 κ-(A)dS quantum algebra, and quantize the resulting contracted Poisson–Hopf algebras, thus giving rise to the κ-deformation of the Newtonian (Newton–Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding κ-Newtonian and κ-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the κ-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter κ, the curvature parameter η and the speed of light parameter c.

Keywords

• Quantum groups
• Carroll
• Newton–Hooke
• Anti-de Sitter
• Kappa-deformation
• Noncommutative spaces