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

In this new paper (arXiv:1408.3689) 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 Λ, 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 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.

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