Contractions, deformations and curvature

TL;DR

This paper explores the geometric interpretation of Lie algebra contractions as zero-curvature limits of homogeneous spaces with constant curvature, and introduces quantum deformations to generate spaces with variable curvature.

Contribution

It provides a geometric framework linking Lie algebra contractions to curvature limits and introduces quantum deformations as a method to create variable curvature spaces.

Findings

Contractions correspond to zero-curvature limits of homogeneous spaces.

Quantum deformations control the curvature of associated quantum spaces.

Detailed analysis of classical and relativistic spacetimes in this framework.

Abstract

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.