Drinfel'd doubles for (2+1)-gravity

Abstract

All possible Drinfel'd double structures for the anti-de Sitter Lie algebra so(2,2) and de Sitter Lie algebra so(3,1) in (2+1)-dimensions are explicitly constructed and analysed in terms of a kinematical basis adapted to (2+1)-gravity. Each of these structures provides in a canonical way a pairing among the (anti-)de Sitter generators, as well as a specific classical r-matrix, and the cosmological constant is included in them as a deformation parameter. It is shown that four of these structures give rise to a Drinfel'd double structure for the Poincaré algebra iso(2,1) in the limit where the cosmological constant tends to zero. We explain how these Drinfel'd double structures are adapted to (2+1)-gravity, and we show that the associated quantum groups are natural candidates for the quantum group symmetries of quantised (2+1)-gravity models and their associated non-commutative spacetimes.

22 pages, no figures

References in corpus (5)

• 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
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Canonical and Lie-algebraic twist deformations of $\kappa$-Poincare and contractions to $\kappa$-Galilei algebras

Cited by in corpus (21)

• The kappa-(A)dS quantum algebra in (3+1) dimensions
• Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting
• On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
• 2+1D Loop Quantum Gravity on the Edge
• 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
• Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
• 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
• Extending general covariance: Moyal-type noncommutative manifolds
• From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes
• Deformed phase spaces with group valued momenta
• Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries
• Quantum symmetries in 2+1 dimensions: Carroll, (a)dS-Carroll, Galilei and (a)dS-Galilei
• Drinfel'd double structures for Poincaré and Euclidean groups
• 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