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

TL;DR

This paper explores the non-relativistic limit of (2+1) gravity, deriving quantum group structures and noncommutative spacetimes from (anti)-de Sitter algebras via Drinfel'd doubles and contraction methods.

Contribution

It introduces a new contraction approach from (anti)-de Sitter quantum doubles to Galilean and Newton-Hooke quantum groups, elucidating their noncommutative spacetime structures.

Findings

Quantum group structures for Galilean and Newton-Hooke algebras are explicitly constructed.

Noncommutative spacetimes feature a commuting time coordinate and noncommuting space coordinates.

The non-relativistic limit of (2+1) gravity is consistent with the Chern-Simons formulation.

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.