Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups

Abstract

The complete classification of classical $r$-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincaré groups such that their Lorentz sector is a quantum subgroup is presented. It is found that there exists three classes of such $r$-matrices, one of them being a novel two-parametric one. The (A)dS and Minkowskian Poisson homogeneous spaces corresponding to these three deformations are explicitly constructed in both local and ambient coordinates. Their quantization is performed, thus giving rise to the associated noncommutative spacetimes, that in the Minkowski case are naturally expressed in terms of quantum null-plane coordinates, and they are always defined by homogeneous quadratic algebras. Finally, non-relativistic and ultra-relativistic limits giving rise to novel Newtonian and Carrollian noncommutative spacetimes are also presented.

30 pages; v2 matches version accepted by Class. Quant. Grav. Comments and references added. The (2+1)-dimensional case is described in new section 4.1

References in corpus (13)

• Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time
• Dynamics of Carroll Particles
• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Three Dimensional Quantum Geometry and Deformed Poincare Symmetry
• Planar Carrollean dynamics, and the Carroll quantum equation
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• The $\kappa$-(A)dS noncommutative spacetime
• $\kappa$-Deformations and Extended $\kappa$-Minkowski Spacetimes
• Deformed Carroll particle from 2+1 gravity
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras
• (Anti)de Sitter/Poincare symmetries and representations from Poincare/Galilei through a classical deformation approach
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Cited by in corpus (1)

• Interplay between spacetime curvature, speed of light and quantum deformations of relativistic symmetries