Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations

TL;DR

This paper demonstrates how 3d gravity spacetimes and their symmetries can be derived from a single quantum deformation of the Lorentz algebra, unifying various geometries through algebraic structures.

Contribution

It introduces a unified quantum deformation framework that generates all constant curvature 3d spacetimes and their symmetries from a single algebraic model.

Findings

Derivation of anti-de Sitter algebra from quantum deformation

Extension to de Sitter, spherical, and hyperbolic spaces via analytic continuation

Recovery of flat and Newtonian spacetimes in specific limits

Abstract

We show how the constant curvature spacetimes of 3d gravity and the associated symmetry algebras can be derived from a single quantum deformation of the 3d Lorentz algebra sl(2,R). We investigate the classical Drinfel'd double of a "hybrid" deformation of sl(2,R) that depends on two parameters (\eta,z). With an appropriate choice of basis and real structure, this Drinfel'd double agrees with the 3d anti-de Sitter algebra. The deformation parameter \eta is related to the cosmological constant, while z is identified with the inverse of the speed of light and defines the signature of the metric. We generalise this result to de Sitter space, the three-sphere and 3d hyperbolic space through analytic continuation in \eta and z; we also investigate the limits of vanishing \eta and z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively.