Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
arXiv:1001.4228 · doi:10.1016/j.physletb.2010.03.043
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 (η,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 ηis 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 ηand z; we also investigate the limits of vanishing ηand z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively.
12 pages; minor changes, additional references
References in corpus (6)
- kappa-Minkowski spacetime as the result of Jordanian twist deformation
- Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
- Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
- A Immirzi-like parameter for 3d quantum gravity
- Differential structure on the -Minkowski spacetime from twist
- Contractions, deformations and curvature