2010/01/31 by Ángel Ballesteros, Angel Ballesteros, Francisco J. Herranz +1
Mathematics · Physics and Astronomy · #Anti-de Sitter space #Black Holes and Theoretical Physics #Constant curvature #Cosmological constant #Cosmology and Gravitation Theories #Curvature #De Sitter space #De Sitter universe #Geometry #Homogeneous space #Lorentz group #Lorentz transformation #Mathematical physics #Mathematics #Minkowski space #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Physics #Quantum #Quantum gravity #Quantum mechanics #Universe #gr-qc #math-ph #math.MP
paper · pdf · doi:10.1016/j.physletb.2010.03.043
published as Phys.Lett.B687:375-381,2010 · 12 pages; minor changes, additional references
arxiv created 2010/03/01 · openalex publication_date 2010/03/18 · arxiv updated 2010/04/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 η 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 η and z; we also investigate the limits of vanishing η and z, which yield the flat spacetimes (Minkowski and Euclidean spaces) and Newtonian models, respectively.