# Cayley-Klein Poisson homogeneous spaces

## TL;DR

This paper reviews the nine two-dimensional Cayley-Klein geometries using graded contractions, constructs new Poisson homogeneous spaces with Poisson-Lie structures, and explores their quantization into noncommutative geometries, highlighting their kinematical interpretations.

## Contribution

It introduces a systematic approach to construct and quantize Cayley-Klein geometries via Poisson-Lie structures, providing new noncommutative models.

## Key findings

- Construction of new Poisson homogeneous spaces for Cayley-Klein geometries
- Quantization yields noncommutative analogues of classical geometries
- Kinematical interpretations relate geometries to spacetime models

## Abstract

The nine two-dimensional Cayley-Klein geometries are firstly reviewed by following a graded contraction approach. Each geometry is considered as a set of three symmetrical homogeneous spaces (of points and two kinds of lines), in such a manner that the graded contraction parameters determine their curvature and signature. Secondly, new Poisson homogeneous spaces are constructed by making use of certain Poisson-Lie structures on the corresponding motion groups. Therefore, the quantization of these spaces provides noncommutative analogues of the Cayley-Klein geometries. The kinematical interpretation for the semi-Riemannian and pseudo-Riemannian Cayley-Klein geometries is emphasized, since they are just Newtonian and Lorentzian spacetimes of constant curvature.