2014/08/31 by Ángel Ballesteros, Angel Ballesteros, Francisco J. Herranz +2
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Canonical quantization #Classical limit #Commutator #Galilean #Group (periodic table) #Homogeneous space #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Quantization (signal processing) #Quantum #Quantum spacetime #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/0264-9381/31/24/245013
published as Class. Quantum Grav. 31 (2014) 245013 · 23 pages. Some comments and references added
arxiv created 2014/11/12 · openalex publication_date 2014/11/28 · arxiv updated 2014/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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.