The 𝜅-Newtonian and 𝜅-Carrollian algebras and their noncommutative spacetimes
Angel Ballesteros, Giulia Gubitosi, Ivan Gutierrez-Sagredo and Francisco J. Herranz
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.