3d gravity and quantum deformations: a Drinfel'd double approach

Abstract

The constant curvature spacetimes of 3d gravity and their associated symmetry algebras are shown to arise from the 6d Drinfel'd double that underlies the two-parametric 'hybrid' quantum deformation of the \U0001d530\U0001d529(2, ℛ) algebra. Moreover, the quantum deformation supplies the additional structures (star structure and pairing) that enter in the Chern-Simons formulation of the theory, thus establishing a direct link between quantum \U0001d530\U0001d529(2, ℛ) algebras and 3d gravity models. In this approach the flat spacetimes and Newtonian models arise as Lie algebra contractions that are governed by two dimensionful \U0001d530\U0001d529(2, ℛ) deformation parameters, which are directly related to the cosmological constant and to the speed of light.