Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

Abstract

We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant $\Lambda$ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like $\kappa$-AdS and dS quantum algebras; their flat limit $\Lambda \to 0$ leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear $\Lambda$-deformation of the $\kappa$-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

References in corpus (8)

• kappa-Minkowski spacetime as the result of Jordanian twist deformation
• Turaev-Viro invariants as an extended TQFT
• 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
• $\kappa$-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant

Cited by in corpus (12)

• The $\kappa$-(A)dS noncommutative spacetime
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• $\kappa$-Deformations and Extended $\kappa$-Minkowski Spacetimes
• 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
• Quantum $D=3$ Euclidean and Poincaré symmetries from contraction limits
• The Poincaré group as a Drinfel'd double
• $\kappa$-de Sitter and $\kappa$-Poincaré symmetries emerging from Chern-Simons (2+1)D gravity with a cosmological constant
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes
• Quantum Klein Space and Superspace
• Noncommutative (A)dS and Minkowski spacetimes from quantum Lorentz subgroups