On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions

Abstract

Quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions are revisited, and several features of these quantum structures are reviewed. In particular, the classification problem of (2+1) (A)dS Lie bialgebras is presented and the associated noncommutative quantum (A)dS spaces are also analysed. Moreover, the flat limit (or vanishing cosmological constant) of all these structures leading to (2+1) quantum Poincaré algebras and groups is simultaneously given by considering the cosmological constant as an explicit Lie algebra parameter in the (A)dS algebras. By making use of this classification, a three-parameter generalization of the κ-deformation for the (2+1) (A)dS algebras and quantum spacetimes is given. Finally, the same problem is studied in (3+1) dimensions, where a two-parameter generalization of the κ-(A)dS deformation that preserves the space isotropy is found.

15 pages, contribution presented at the Conference "3Quantum: Algebra, Geometry, Information", Tallinn (Estonia), July 2012. Minor corrections, one reference added

References in corpus (6)

• 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
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Drinfel'd doubles for (2+1)-gravity
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras
• Classification of real three-dimensional Poisson-Lie groups

Cited by in corpus (20)

• Hamilton geometry: Phase space geometry from modified dispersion relations
• Planck-scale-modified dispersion relations in homogeneous and isotropic spacetimes
• The kappa-(A)dS quantum algebra in (3+1) dimensions
• The $\kappa$-(A)dS noncommutative spacetime
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• Kinematics of particles with quantum de Sitter symmetries
• Curved momentum spaces from quantum groups with cosmological constant
• Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes
• Quantum deformations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries in unified $\mathfrak{o}(4;\mathbb{C})$ setting -- Addendum
• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• AdS Poisson homogeneous spaces and Drinfel'd doubles
• Quantum Twist-Deformed D=4 Phase Spaces with Spin Sector and Hopf Algebroid Structures
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• The Poincaré group as a Drinfel'd double
• Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces
• Quantizations of D=3 Lorentz symmetry
• Quantum groups and noncommutative spacetimes with cosmological constant
• Darboux families and the classification of real four-dimensional indecomposable coboundary Lie bialgebras
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups