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

被引:7
作者
Ballesteros, Angel [1 ]
Herranz, Francisco J. [1 ]
Naranjo, Pedro [1 ]
机构
[1] Univ Burgos, Dept Fis, E-09001 Burgos, Spain
关键词
(2+1) gravity; noncommutative spacetime; non-relativistic limit; Galilean spacetimes; cosmological constant; quantum groups; Poisson-Lie groups; CHERN-SIMONS THEORY; (2+1)-DIMENSIONAL GRAVITY; COSMOLOGICAL CONSTANT; SPECIAL RELATIVITY; QUANTUM-MECHANICS; SYMMETRY; ALGEBRAS; DEFORMATION; SPIN;
D O I
10.1088/0264-9381/31/24/245013
中图分类号
P1 [天文学];
学科分类号
0704 ;
摘要
It is shown that the canonical classical r-matrix arising from the Drinfel'd 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., 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 (x) over cap (0) together with two noncommutative space coordinates ((x) over cap (1), (x) over cap (2)), whose commutator is a function of the cosmological constant. and of the (central) 'quantum time' coordinate (x) over cap (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.
引用
收藏
页数:25
相关论文
共 48 条
[1]   A CHERN-SIMONS ACTION FOR 3-DIMENSIONAL ANTI-DESITTER SUPERGRAVITY THEORIES [J].
ACHUCARRO, A ;
TOWNSEND, PK .
PHYSICS LETTERS B, 1986, 180 (1-2) :89-92
[2]   Non-relativistic spacetimes with cosmological constant [J].
Aldrovandi, R ;
Barbosa, AL ;
Crispino, LCB ;
Pereira, JG .
CLASSICAL AND QUANTUM GRAVITY, 1999, 16 (02) :495-506
[3]   SYMPLECTIC STRUCTURE OF THE MODULI SPACE OF FLAT CONNECTION ON A RIEMANN SURFACE [J].
ALEKSEEV, AY ;
MALKIN, AZ .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1995, 169 (01) :99-119
[4]   COMBINATORIAL QUANTIZATION OF THE HAMILTONIAN CHERN-SIMONS THEORY .1. [J].
ALEKSEEV, AY ;
GROSSE, H ;
SCHOMERUS, V .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1995, 172 (02) :317-358
[5]   Representation theory of Chern-Simons observables [J].
Alekseev, AY ;
Schomerus, V .
DUKE MATHEMATICAL JOURNAL, 1996, 85 (02) :447-510
[6]   Quantum symmetry, the cosmological constant and Planck-scale phenomenology [J].
Amelino-Camelia, G ;
Smolin, L ;
Starodubtsev, A .
CLASSICAL AND QUANTUM GRAVITY, 2004, 21 (13) :3095-3110
[7]   Doubly-Special Relativity: Facts, Myths and Some Key Open Issues [J].
Amelino-Camelia, Giovanni .
SYMMETRY-BASEL, 2010, 2 (01) :230-271
[8]   POSSIBLE KINEMATICS [J].
BACRY, H ;
LEVYLEBL.JM .
JOURNAL OF MATHEMATICAL PHYSICS, 1968, 9 (10) :1605-&
[9]   QUANTUM (2+1) KINEMATICAL ALGEBRAS - A GLOBAL APPROACH [J].
BALLESTEROS, A ;
HERRANZ, FJ ;
DELOLMO, MA ;
SANTANDER, M .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1994, 27 (04) :1283-1297
[10]   QUANTUM STRUCTURE OF THE MOTION GROUPS OF THE 2-DIMENSIONAL CAYLEY-KLEIN GEOMETRIES [J].
BALLESTEROS, A ;
HERRANZ, FJ ;
DELOLMO, MA ;
SANTANDER, M .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1993, 26 (21) :5801-5823