From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes

Abstract

It is shown that the canonical classical $r$-matrix arising from the Drinfel'd double structure underlying the two-fold centrally extended (2+1) Galilean and Newton-Hooke Lie algebras (with either zero or non-zero cosmological constant $\Lambda$, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical $r$-matrices associated with the Drinfel'd double structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and Newton-Hooke Drinfel'd doubles is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting 'absolute time' coordinate ${\hat{x}}_0$ together with two noncommutative space coordinates $({\hat{x}}_1,{\hat{x}}_2)$, whose commutator is a function of the cosmological constant $\Lambda$ and of the (central) 'quantum time' coordinate ${\hat{x}}_0$. Thus, the Chern-Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

arXiv comment23 pages. Some comments and references added

References in corpus (31)

• A WZW model based on a non-semi-simple group
• Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness
• The exotic Galilei group and the "Peierls substitution"
• Quantum symmetry, the cosmological constant and Planck scale phenomenology
• Galilean-Invariant (2+1)-Dimensional Models with a Chern-Simons-Like Term and D=2 N oncommutative Geometry
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity
• Doubly Special Relativity and de Sitter space
• Symplectic structure of the moduli space of flat connections on a Riemann surface
• Newton-Hooke spacetimes, Hpp-waves and the cosmological constant
• Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Two dimensional lattice gauge theory based on a quantum group
• (2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase
• Anyon spin and the exotic central extension of the planar Galilei group
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane
• A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
• Non-Relativistic Spacetimes with Cosmological Constant
• Poisson structure and symmetry in the Chern-Simons formulation of (2+1)-dimensional gravity
• Conformal compactification and cycle-preserving symmetries of spacetimes
• Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
• Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
• Spin and exotic Galilean symmetry
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations
• Drinfel'd doubles for (2+1)-gravity
• A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant
• Central Extensions of the Quasi-orthogonal Lie algebras
• Two particle Quantummechanics in 2+1 Gravity using Non Commuting Coordinates
• Twisted (2+1) $\kappa$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes
• 2+1 Kinematical expansions: from Galilei to de Sitter algebras

Cited by in corpus (5)

• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• Lorentzian Snyder spacetimes and their Galilei and Carroll limits from projective geometry
• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications
• The Poincaré group as a Drinfel'd double