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.

23 pages. Some comments and references added

References in corpus (6)

• (2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions
• 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
• Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations

Cited by in corpus (2)

• Towards (3+1) gravity through Drinfel'd doubles with cosmological constant
• Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications