Cayley-Klein Poisson homogeneous spaces
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.
Cite
@article{arxiv.1812.11883,
title = {Cayley-Klein Poisson homogeneous spaces},
author = {Francisco J. Herranz and Angel Ballesteros and Ivan Gutierrez-Sagredo and Mariano Santander},
journal= {arXiv preprint arXiv:1812.11883},
year = {2019}
}
Comments
22 pages. Based on the contribution presented at the "XXth International Conference on Geometry, Integrability and Quantization" held in Varna, Bulgaria, June 2-7, 2018