From Lorentzian to Galilean (2+1) gravity: Drinfelʼd doubles, quantization and noncommutative spacetimes

Abstract

It is shown that the canonical classical r-matrix arising from the Drinfelʼd double (DD) structure underlying the two-fold centrally extended (2+1) Galilean and Newton–Hooke (NH) Lie algebras (with either zero or non-zero cosmological constant Λ, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical r-matrices associated with the DD structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and NH DD is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting ‘absolute time’ coordinate together with two noncommutative space coordinates , whose commutator is a function of the cosmological constant Λ and of the (central) ‘quantum time’ coordinate . 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.