Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 052, 26 pages      arXiv:1403.4773      http://dx.doi.org/10.3842/SIGMA.2014.052
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

Twisted (2+1) κ-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

Ángel Ballesteros a, Francisco J. Herranz a, Catherine Meusburger b and Pedro Naranjo a
a) Departamento de Física, Universidad de Burgos, E-09001 Burgos, Spain
b) Department Mathematik, Friedrich-Alexander Universität Erlangen-Nürnberg, Cauerstr. 11, D-91058 Erlangen, Germany

Received March 09, 2014, in final form May 13, 2014; Published online May 18, 2014

Abstract
We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant Λ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like κ-AdS and dS quantum algebras; their flat limit Λ→0 leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear Λ-deformation of the κ-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

Key words: (2+1)-gravity; deformation; non-commutative spacetime; anti-de Sitter; cosmological constant; quantum groups; Poisson-Lie groups; contraction.

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References

  1. Amelino-Camelia G., Doubly-special relativity: first results and key open problems, Internat. J. Modern Phys. D 11 (2002), 1643-1669, gr-qc/0210063.
  2. Amelino-Camelia G., Quantum-spacetime phenomenology, Living Rev. Relativity 16 (2013), 5, 137 pages, arXiv:0806.0339.
  3. Amelino-Camelia G., Smolin L., Starodubtsev A., Quantum symmetry, the cosmological constant and Planck-scale phenomenology, Classical Quantum Gravity 21 (2004), 3095-3110, hep-th/0306134.
  4. Arratia O., Herranz F.J., del Olmo M.A., Bicrossproduct structure of the null-plane quantum Poincaré algebra, J. Phys. A: Math. Gen. 31 (1998), L1-L7, q-alg/9707025.
  5. Bacry H., Lévy-Leblond J.M., Possible kinematics, J. Math. Phys. 9 (1968), 1605-1614.
  6. Bais F.A., Müller N.M., Topological field theory and the quantum double of SU(2), Nuclear Phys. B 530 (1998), 349-400, hep-th/9804130.
  7. Bais F.A., Müller N.M., Schroers B.J., Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity, Nuclear Phys. B 640 (2002), 3-45, hep-th/0205021.
  8. Ballesteros Á., Bruno N.R., Herranz F.J., A new `doubly special relativity' theory from a quantum Weyl-Poincaré algebra, J. Phys. A: Math. Gen. 36 (2003), 10493-10503, hep-th/0305033.
  9. Ballesteros Á., Bruno N.R., Herranz F.J., A non-commutative Minkowskian spacetime from a quantum AdS algebra, Phys. Lett. B 574 (2003), 276-282, hep-th/0306089.
  10. Ballesteros Á., Bruno N.R., Herranz F.J., Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti)de Sitter groups, hep-th/0401244.
  11. Ballesteros Á., Herranz F.J., del Olmo M.A., Santander M., Four-dimensional quantum affine algebras and space-time q-symmetries, J. Math. Phys. 35 (1994), 4928-4940, hep-th/9310140.
  12. Ballesteros Á., Herranz F.J., del Olmo M.A., Santander M., Quantum (2+1) kinematical algebras: a global approach, J. Phys. A: Math. Gen. 27 (1994), 1283-1297.
  13. Ballesteros Á., Herranz F.J., del Olmo M.A., Santander M., A new "null-plane" quantum Poincaré algebra, Phys. Lett. B 351 (1995), 137-145, q-alg/9502019.
  14. Ballesteros Á., Herranz F.J., del Olmo M.A., Santander M., Non-standard quantum so(2,2) and beyond, J. Phys. A: Math. Gen. 28 (1995), 941-955, hep-th/9406098.
  15. Ballesteros Á., Herranz F.J., Meusburger C., Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations, Phys. Lett. B 687 (2010), 375-381, arXiv:1001.4228.
  16. Ballesteros Á., Herranz F.J., Meusburger C., Drinfel'd doubles for (2+1)-gravity, Classical Quantum Gravity 30 (2013), 155012, 20 pages, arXiv:1303.3080.
  17. Ballesteros Á., Herranz F.J., Meusburger C., A (2+1) non-commutative Drinfel'd double spacetime with cosmological constant, Phys. Lett. B 732 (2014), 201-209, arXiv:1402.2884.
  18. Ballesteros Á., Herranz F.J., Musso F., On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions, J. Phys. Conf. Ser., to appear, arXiv:1302.0684.
  19. Ballesteros Á., Herranz F.J., Pereña C.M., Null-plane quantum universal R-matrix, Phys. Lett. B 391 (1997), 71-77, q-alg/9607009.
  20. Batista E., Majid S., Noncommutative geometry of angular momentum space U(su(2)), J. Math. Phys. 44 (2003), 107-137, hep-th/0205128.
  21. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  22. Borowiec A., Pachoł A., κ-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  23. Borowiec A., Pachoł A., κ-Minkowski spacetimes and DSR algebras: fresh look and old problems, SIGMA 6 (2010), 086, 31 pages, arXiv:1005.4429.
  24. Bruno N.R., Amelino-Camelia G., Kowalski-Glikman J., Deformed boost transformations that saturate at the Planck scale, Phys. Lett. B 522 (2001), 133-138, hep-th/0107039.
  25. Celeghini E., Giachetti R., Sorace E., Tarlini M., The three-dimensional Euclidean quantum group E(3)q and its R-matrix, J. Math. Phys. 32 (1991), 1159-1165.
  26. Celeghini E., Giachetti R., Sorace E., Tarlini M., Contractions of quantum groups, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 221-244.
  27. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  28. Daszkiewicz M., Canonical and Lie-algebraic twist deformations of κ-Poincaré and contractions to κ-Galilei algebras, Internat. J. Modern Phys. A 23 (2008), 4387-4400, arXiv:0807.0133.
  29. Daszkiewicz M., Generalized twist deformations of Poincaré and Galilei Hopf algebras, Rep. Math. Phys. 63 (2009), 263-277, arXiv:0812.1613.
  30. Daszkiewicz M., Twist deformations of Newton-Hooke Hopf algebras, Modern Phys. Lett. A 24 (2009), 1325-1334, arXiv:0904.0432.
  31. de Azcárraga J.A., del Olmo M.A., Pérez Bueno J.C., Santander M., Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras, J. Phys. A: Math. Gen. 30 (1997), 3069-3086, q-alg/9612022.
  32. de Azcárraga J.A., Pérez Bueno J.C., Deformed and extended Galilei group Hopf algebras, J. Phys. A: Math. Gen. 29 (1996), 6353-6362, q-alg/9507005.
  33. Drinfel'd V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1983), 68-71.
  34. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  35. Freidel L., Kowalski-Glikman J., Smolin L., 2+1 gravity and doubly special relativity, Phys. Rev. D 69 (2004), 044001, 7 pages, hep-th/0307085.
  36. Garay L.J., Quantum gravity and minimum length, Internat. J. Modern Phys. A 10 (1995), 145-165, gr-qc/9403008.
  37. Gomez X., Classification of three-dimensional Lie bialgebras, J. Math. Phys. 41 (2000), 4939-4956.
  38. Herranz F.J., Ballesteros Á., Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006), 010, 22 pages, math-ph/0512084.
  39. Herranz F.J., de Montigny M., del Olmo M.A., Santander M., Cayley-Klein algebras as graded contractions of so(N+1), J. Phys. A: Math. Gen. 27 (1994), 2515-2526, hep-th/9312126.
  40. Herranz F.J., Santander M., Conformal symmetries of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6601-6618, math-ph/0110019.
  41. Inönü E., Wigner E.P., On the contraction of groups and their representations, Proc. Nat. Acad. Sci. USA 39 (1953), 510-524.
  42. Jimbo M., A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  43. Kirillov Jr. A., Balsam B., Turaev-Viro invariants as an extended TQFT, arXiv:1004.1533.
  44. Kowalski-Glikman J., Nowak S., Doubly special relativity theories as different bases of κ-Poincaré algebra, Phys. Lett. B 539 (2002), 126-132, hep-th/0203040.
  45. Lukierski J., Lyakhovsky V., Mozrzymas M., κ-deformations of D=4 Weyl and conformal symmetries, Phys. Lett. B 538 (2002), 375-384, hep-th/0203182.
  46. Lukierski J., Minnaert P., Mozrzymas M., Quantum deformations of conformal algebras introducing fundamental mass parameters, Phys. Lett. B 371 (1996), 215-222, q-alg/9507005.
  47. Lukierski J., Nowicki A., Doubly special relativity versus κ-deformation of relativistic kinematics, Internat. J. Modern Phys. A 18 (2003), 7-18, hep-th/0203065.
  48. Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and κ-deformed field theory, Phys. Lett. B 293 (1992), 344-352.
  49. Lukierski J., Ruegg H., Quantum κ-Poincaré in any dimension, Phys. Lett. B 329 (1994), 189-194, hep-th/9310117.
  50. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., q-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  51. Lukierski J., Ruegg H., Zakrzewski W.J., Classical and quantum mechanics of free k-relativistic systems, Ann. Physics 243 (1995), 90-116, hep-th/9312153.
  52. Magueijo J., Smolin L., Lorentz invariance with an invariant energy scale, Phys. Rev. Lett. 88 (2002), 190403, 4 pages, hep-th/0112090.
  53. Majid S., Hopf algebras for physics at the Planck scale, Classical Quantum Gravity 5 (1988), 1587-1606.
  54. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  55. Majid S., Ruegg H., Bicrossproduct structure of κ-Poincaré group and non-commutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9405107.
  56. Marcianò A., Amelino-Camelia G., Bruno N.R., Gubitosi G., Mandanici G., Melchiorri A., Interplay between curvature and Planck-scale effects in astrophysics and cosmology, J. Cosmol. Astropart. Phys. 2010 (2010), no. 6, 030, 29 pages, arXiv:1004.1110.
  57. Maślanka P., The n-dimensional κ-Poincaré algebra and group, J. Phys. A: Math. Gen. 26 (1993), L1251-L1253.
  58. Meusburger C., Schroers B.J., Quaternionic and Poisson-Lie structures in three-dimensional gravity: the cosmological constant as deformation parameter, J. Math. Phys. 49 (2008), 083510, 27 pages, arXiv:0708.1507.
  59. Meusburger C., Schroers B.J., Generalised Chern-Simons actions for 3d gravity and κ-Poincaré symmetry, Nuclear Phys. B 806 (2009), 462-488, arXiv:0805.3318.
  60. Šnobl L., Hlavatý L., Classification of six-dimensional real Drinfeld doubles, Internat. J. Modern Phys. A 17 (2002), 4043-4067, math.QA/0202210.
  61. Takhtajan L.A., Lectures on quantum groups, in Introduction to Quantum Group and Integrable Massive Models of Quantum Field Theory (Nankai, 1989), Nankai Lectures Math. Phys., World Sci. Publ., River Edge, NJ, 1990, 69-197.
  62. Turaev V., Virelizier A., On two approaches to 3-dimensional TQFTs, arXiv:1006.3501.
  63. Witten E., 2+1-dimensional gravity as an exactly soluble system, Nuclear Phys. B 311 (1988), 46-78.
  64. Zakrzewski S., Quantum Poincaré group related to the κ-Poincaré algebra, J. Phys. A: Math. Gen. 27 (1994), 2075-2082.
  65. Zakrzewski S., Poisson structures on Poincaré group, Comm. Math. Phys. 185 (1997), 285-311, q-alg/9602001.

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