A general approach to noncommutative spaces from Poisson homogeneous spaces: Applications to (A)dS and Poincar\'e

TL;DR

This paper introduces a general method to construct noncommutative spaces with quantum group symmetry from Poisson homogeneous spaces, explicitly applying it to (A)dS and Poincaré spacetimes, including $ ext{kappa}$-deformations.

Contribution

It provides a unified framework for quantizing coisotropic Poisson homogeneous spaces associated with coboundary Lie bialgebras, with explicit examples for (3+1)D (A)dS and Poincaré cases.

Findings

Explicit forms of noncommutative (A)dS and Poincaré spaces derived.

Construction of all noncommutative Minkowski and (A)dS spacetimes with quantum Lorentz symmetry.

Identification of three 6D $ ext{kappa}$-Poincaré spaces of time-like type.

Abstract

In this contribution we present a general procedure that allows the construction of noncommutative spaces with quantum group invariance as the quantization of their associated coisotropic Poisson homogeneous spaces coming from a coboundary Lie bialgebra structure. The approach is illustrated by obtaining in an explicit form several noncommutative spaces from (3+1)D (A)dS and Poincar\'e coisotropic Lie bialgebras. In particular, we review the construction of the $\kappa$-Minkowski and $\kappa$-(A)dS spacetimes in terms of the cosmological constant $\Lambda$. Furthermore, we present all noncommutative Minkowski and (A)dS spacetimes that preserved a quantum Lorentz subgroup. Finally, it is also shown that the same setting can be used to construct the three possible 6D $\kappa$-Poincar\'e spaces of time-like. Some open problems are also addressed.