Curvature from quantum deformations

TL;DR

This paper demonstrates how a Poisson coalgebra analogue of quantum deformation of sl(2) generates integrable and superintegrable geodesic dynamics on 2D spaces with variable or constant curvature, linking quantum algebra to geometric properties.

Contribution

It introduces a novel connection between quantum deformations and geometric dynamics, enabling the construction of integrable systems on curved spaces with variable or constant curvature.

Findings

Quantum deformation parameter z controls space curvature.

Flat space recovered as z approaches zero.

Spaces include Riemannian and relativistic spacetimes with constant curvature.

Abstract

A Poisson coalgebra analogue of a (non-standard) quantum deformation of sl(2) is shown to generate an integrable geodesic dynamics on certain 2D spaces of non-constant curvature. Such a curvature depends on the quantum deformation parameter z and the flat case is recovered in the limit z\to 0. A superintegrable geodesic dynamics can also be defined in the same framework, and the corresponding spaces turn out to be either Riemannian or relativistic spacetimes (AdS and dS) with constant curvature equal to z. The underlying coalgebra symmetry of this approach ensures the existence of its generalization to arbitrary dimension.