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

TL;DR

This paper classifies all classical r-matrices leading to quantum deformations of (A)dS and Poincaré groups with quantum Lorentz subgroups, constructs their associated noncommutative spacetimes, and explores non-relativistic limits.

Contribution

It provides a complete classification of r-matrices for quantum (A)dS and Poincaré groups with quantum Lorentz subgroups, including a new two-parametric class, and constructs their noncommutative spacetimes.

Findings

Identified three classes of r-matrices, including a novel two-parametric one.

Explicitly constructed noncommutative (A)dS and Minkowski spacetimes.

Derived non-relativistic and ultra-relativistic limits leading to new Newtonian and Carrollian spacetimes.

Abstract

The complete classification of classical $r$-matrices generating quantum deformations of the (3+1)-dimensional (A)dS and Poincar\'e 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.