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Advances in High Energy Physics
Volume 2017, Article ID 7876942, 19 pages
https://doi.org/10.1155/2017/7876942
Review Article

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain

Correspondence should be addressed to Francisco J. Herranz; se.ubu@znarrehjf

Received 12 May 2017; Accepted 6 August 2017; Published 28 November 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 Ángel Ballesteros et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Linked References

  1. S. Majid, “Hopf algebras for physics at the Planck scale,” Classical and Quantum Gravity, vol. 5, no. 12, pp. 1587–1606, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. P. P. Kulish and N. Yu. Reshetikhin, “Quantum linear problem for the sine-Gordon equation and higher representations,” Proceedings of the Steklov Institute of Mathematics, vol. 101, pp. 101–110, 1981 (Russian), English translation: Journal of Soviet Mathematics, vol. 23, no. 4, pp. 2435–2441, 1983. View at Google Scholar
  3. V. G. Drinfeld, “Quantum groups,” in Proceedings of the International Congress of Mathematicians, A. V. Gleason, Ed., vol. 1, pp. 798–820, AMS, Berkeley, Calif, USA, 1987.
  4. M. Jimbo, “Aq-difference analogue of U(g) and the Yang-Baxter equation,” Letters in Mathematical Physics, vol. 10, no. 1, pp. 63–69, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  5. L. A. Takhtajan, “Lectures on quantum groups,” in Introduction to Quantum Group and Integrable Massive Models of Quantum Field Theory (Nankai, 1989), Nankai Lectures Math. Phys., pp. 69–197, World Scientific, River Edge, NJ, USA, 1990. View at Google Scholar · View at MathSciNet
  6. N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantization of Lie groups and Lie algebras,” Algebra i Analiz, vol. 1, no. 1, pp. 178–206, 1989 (Russian), English translation: Leningrad Mathematical Journal, vol. 1, no. 1, pp. 193–225, 1990. View at Google Scholar
  7. V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, UK, 1994. View at MathSciNet
  8. S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, UK, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. Lukierski, H. Ruegg, A. Nowicki, and V. N. Tolstoy, “q-deformation of Poincaré algebra,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 264, no. 3-4, pp. 331–338, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  10. E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini, “The quantum Heisenberg group ,” Journal of Mathematical Physics, vol. 32, no. 5, pp. 1155–1158, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  11. E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini, “The three-dimensional Euclidean quantum group and its -matrix,” Journal of Mathematical Physics, vol. 32, no. 5, pp. 1159–1165, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  12. F. Bonechi, E. Celeghini, R. Giachetti, E. Sorace, and M. Tarlini, “Inhomogeneous quantum groups as symmetries of phonons,” Physical Review Letters, vol. 68, no. 25, pp. 3718–3720, 1992. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Giller, P. Kosinski, M. Majewski, P. Maslanka, and J. Kunz, “More about the -deformed Poincaré algebra,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 286, no. 1-2, pp. 57–62, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. Lukierski, A. Nowicki, and H. Ruegg, “New quantum Poincaré algebra and k-deformed field theory,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 293, no. 3-4, pp. 344–352, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  15. P. Maslanka, “The n-dimensional kappa -Poincaré algebra and group,” Journal of Physics. A. Mathematical and General, vol. 26, no. 24, pp. L1251–L1253, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  16. S. Majid and H. Ruegg, “Bicrossproduct structure of k-Poincaré group and non-commutative geometry,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 334, no. 3-4, pp. 348–354, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Zakrzewski, “Quantum Poincaré group related to the kappa -Poincaré algebra,” Journal of Physics. A. Mathematical and General, vol. 27, no. 6, pp. 2075–2082, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. Lukierski and H. Ruegg, “Quantum -Poincaré in any dimension,” Physics Letters B, vol. 329, no. 2-3, pp. 189–194, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santander, “Quantum kinematical algebras: a global approach,” Journal of Physics. A. Mathematical and General, vol. 27, no. 4, pp. 1283–1297, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  20. A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santander, “Four-dimensional quantum affine algebras and space-time q-symmetries,” Journal of Mathematical Physics, vol. 35, no. 9, pp. 4928–4940, 1994. View at Google Scholar
  21. A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santander, “A new “null-plane” quantum Poincaré algebra,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 351, no. 1-3, pp. 137–145, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. Lukierski, H. Ruegg, and W. J. Zakrzewski, “Classical and quantum mechanics of free κ-relativistic systems,” Annals of Physics, vol. 243, no. 1, pp. 90–116, 1995. View at Google Scholar
  23. A. Ballesteros, F. J. Herranz, M. A. del Olmo, C. M. Pereña, and M. Santander, “Non-standard quantum Poincaré group: a -matrix approach,” Journal of Physics. A. Mathematical and General, vol. 28, no. 24, pp. 7113–7125, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  24. G. Amelino-Camelia, “Testable scenario for relativity with minimum length,” Physics Letters B, vol. 510, no. 1–4, pp. 255–263, 2001. View at Publisher · View at Google Scholar
  25. G. Amelino-Camelia, “Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale,” International Journal of Modern Physics D, vol. 11, no. 1, pp. 35–59, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Amelino-Camelia, “Doubly-special relativity: first results and key open problems,” International Journal of Modern Physics D, vol. 11, no. 10, pp. 1643–1669, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. Magueijo and L. Smolin, “Lorentz invariance with an invariant energy scale,” Physical Review Letters, vol. 88, Article ID 190403, 2002. View at Publisher · View at Google Scholar
  28. J. Kowalski-Glikman and S. Nowak, “Doubly special relativity theories as different bases of k-Poincaré algebra,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 539, no. 1-2, pp. 126–132, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  29. J. Kowalski-Glikman, “De Sitter space as an arena for doubly special relativity,” Physics Letters B, vol. 547, no. 3-4, pp. 291–296, 2002. View at Publisher · View at Google Scholar
  30. J. Kowalski-Glikman and S. Nowak, “Doubly special relativity and de Sitter space,” Classical and Quantum Gravity, vol. 20, no. 22, pp. 4799–4816, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. J. Lukierski and A. Nowicki, “Doubly special relativity versus κ-deformation of relativistic kinematics,” International Journal of Modern Physics A, vol. 18, no. 1, pp. 7–18, 2003. View at Google Scholar
  32. Á. Ballesteros, N. Rossano Bruno, and F. J. Herranz, “A new ‘doubly special relativity’ theory from a quantum Weyl–Poincaré algebra,” Journal of Physics. A. Mathematical and General, vol. 36, no. 42, pp. 10493–10503, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  33. J. Polchinski, “TASI lectures on D-branes,” in Proceedings of the TASI 96: Fields, Strings, and Duality, C. Efthimiou and B. Greene, Eds., pp. 293–356, World Scientific, Singapore, 1997, http://xxx.lanl.gov/abs/hep-th/9611050.
  34. S. Carlip, “Quantum gravity: a progress report,” Reports on Progress in Physics, vol. 64, no. 8, pp. 885–942, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  35. L. Smolin, “Quantum gravity with a positive cosmological constant,” https://arxiv.org/abs/hep-th/0209079, In press.
  36. S. Förste, “Strings, branes and extra dimensions,” Fortschritte der Physik. Progress of Physics, vol. 50, no. 3-4, pp. 221–403, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  37. L. Freidel, E. R. Livine, and C. Rovelli, “Spectra of length and area in (2 + 1) Lorentzian loop quantum gravity,” Classical and Quantum Gravity, vol. 20, no. 8, pp. 1463–1478, 2003. View at Publisher · View at Google Scholar · View at Scopus
  38. G. Amelino-Camelia, L. Smolin, and A. Starodubtsev, “Quantum symmetry, the cosmological constant and Planck-scale phenomenology,” Classical and Quantum Gravity, vol. 21, no. 13, pp. 3095–3110, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. L. Freidel, J. Kowalski-Glikman, and L. Smolin, “ gravity and doubly special relativity,” Physical Review. D. Third Series, vol. 69, no. 4, Article ID 044001, 7 pages, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  40. H. S. Snyder, “Quantized space-time,” Physical Review, vol. 71, no. 1, pp. 38–41, 1947. View at Publisher · View at Google Scholar
  41. P. Podleś and S. L. Woronowicz, “Quantum deformation of Lorentz group,” Communications in Mathematical Physics, vol. 130, no. 2, pp. 381–431, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  42. S. Doplicher, K. Fredenhagen, and J. E. Roberts, “The quantum structure of spacetime at the Planck scale and quantum fields,” Communications in Mathematical Physics, vol. 172, no. 1, pp. 187–220, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. R. J. Szabo, “Quantum field theory on noncommutative spaces,” Physics Reports A: Review Section of Physics Letters, vol. 378, no. 4, pp. 207–299, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. J. J. Heckman and H. Verlinde, “Covariant non-commutative space-time,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 894, pp. 58–74, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  45. M. Maggiore, “The algebraic structure of the generalized uncertainty principle,” Physics Letters B, vol. 319, no. 1–3, pp. 83–86, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  46. L. J. Garay, “Quantum gravity and minimum length,” International Journal of Modern Physics A, vol. 10, no. 2, pp. 145–165, 1995. View at Google Scholar
  47. A. Ballesteros, N. R. Bruno, and F. J. Herranz, “A non-commutative Minkowskian spacetime from a quantum AdS algebra,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 574, no. 3-4, pp. 276–282, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  48. A. Marcianò, G. Amelino-Camelia, N. R. Bruno, G. Gubitosi, G. Mandanici, and A. Melchiorri, “Interplay between curvature and Planck-scale effects in astrophysics and cosmology,” Journal of Cosmology and Astroparticle Physics, vol. 2010, no. 06, p. 030, 2010. View at Publisher · View at Google Scholar
  49. G. Amelino-Camelia, L. Barcaroli, DG. Amico, N. Loret, and G. Rosati, “IceCube and GRB neutrinos propagating in quantum spacetime,” Physics Letters B, vol. 761, pp. 318–325, 2016. View at Publisher · View at Google Scholar
  50. G. Amelino-Camelia, G. Gubitosi, and G. Palmisano, “Pathways to relativistic curved momentum spaces: de Sitter case study,” International Journal of Modern Physics. D. Gravitation, Astrophysics, Cosmology, vol. 25, no. 2, Article ID 1650027, 36 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  51. L. Barcaroli and G. Gubitosi, “Kinematics of particles with quantum-de Sitter-inspired symmetries,” Physical Review D, vol. 93, no. 12, Article ID 124063, 2016. View at Google Scholar
  52. A. Ballesteros, F. J. Herranz, O. Ragnisco, and M. Santander, “Contractions, deformations and curvature,” International Journal of Theoretical Physics, vol. 47, no. 3, pp. 649–663, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  53. F. J. Herranz and M. Santander, “(Anti)de Sitter/Poincaré symmetries and representations from Poincaré/Galilei through a classical deformation approach,” Journal of Physics. A. Mathematical and Theoretical, vol. 41, no. 1, Article ID 015204, p. 16, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  54. V. G. Drinfel’d, “On Poisson homogeneous spaces of Poisson-Lie groups,” Theoretical and Mathematical Physics, vol. 95, no. 2, pp. 524-525, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  55. M. S. Dijkhuizen and T. H. Koornwinder, “Quantum homogeneous spaces, duality and quantum -spheres,” Geometriae Dedicata, vol. 52, no. 3, pp. 291–315, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  56. S. Zakrzewszki, “Poisson homogeneous spaces,” in Quantum Groups, Formalism and Applications, J. Lukierski, Z. Popowicz, and J. Sobczyk, Eds., pp. 629–639, PWN, Warszaw, Poland, 1995. View at Google Scholar
  57. A. G. Reyman, “Poisson structures related to quantum groups,” in Quantum Groups and Their Applications in Physics, L. Castellani and J. Wess, Eds., pp. 407–444, IOS Press, Amsterdam, The Netherlands, 1996. View at Google Scholar
  58. N. Ciccoli, “Quantum planes and quantum cylinders from Poisson homogeneous spaces,” Journal of Physics. A. Mathematical and General, vol. 29, no. 7, pp. 1487–1495, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  59. S. Majid and B. J. Schroers, “-deformation and semidualization in 3D quantum gravity,” Journal of Physics. A. Mathematical and Theoretical, vol. 42, no. 42, Article ID 425402, p. 40, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  60. P. K. Osei and B. J. Schroers, “On the semiduals of local isometry groups in three-dimensional gravity,” Journal of Mathematical Physics, vol. 53, no. 7, Article ID 073510, p. 26, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  61. P. K. Osei and B. J. Schroers, “Classical -matrices via semidualisation,” Journal of Mathematical Physics, vol. 54, no. 10, Article ID 101702, p. 17, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  62. E. Inonu and E. P. Wigner, “On the contraction of groups and their representations,” Proceedings of the National Academy of Sciences of the United States of America, vol. 39, no. 6, pp. 510–524, 1953. View at Publisher · View at Google Scholar
  63. H. Bacry and J.-M. Lévy-Leblond, “Possible kinematics,” Journal of Mathematical Physics, vol. 9, pp. 1605–1614, 1968. View at Publisher · View at Google Scholar · View at MathSciNet
  64. F. J. Herranz and M. Santander, “The general solution of the real graded contractions of ,” Journal of Physics. A. Mathematical and General, vol. 29, no. 20, pp. 6643–6652, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  65. R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, Wiley, New York, NY, USA, 1994. View at MathSciNet
  66. F. J. Herranz and M. Santander, “Homogeneous phase spaces: the Cayley-Klein framework,” in Geometria y Fisica, J. F. Cariñena, E. Martinez, and M. F. Rañada, Eds., vol. XXXII of Memorias de la Real Academia de Ciencias, pp. 59–84, Real Academia de Ciencias, Madrid, Spain, 1998, https://arxiv.org/abs/physics/9702030. View at Google Scholar
  67. F. J. Herranz and M. Santander, “Conformal symmetries of spacetimes,” Journal of Physics. A. Mathematical and General, vol. 35, no. 31, pp. 6601–6618, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  68. A. Ballesteros, N. A. Gromov, F. J. Herranz, M. A. del Olmo, and M. Santander, “Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras,” Journal of Mathematical Physics, vol. 36, no. 10, pp. 5916–5937, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  69. R. Aldrovandi, A. L. Barbosa, L. C. Crispino, and J. G. Pereira, “Non-relativistic spacetimes with cosmological constant,” Classical and Quantum Gravity, vol. 16, no. 2, pp. 495–506, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  70. V. G. Drinfel’d, “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations,” Soviet Mathematics Doklady, vol. 27, pp. 68–71, 1983. View at Google Scholar
  71. N. R. Bruno, G. Amelino-Camelia, and J. Kowalski-Glikman, “Deformed boost transformations that saturate at the Planck scale,” Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, vol. 522, no. 1-2, pp. 133–138, 2001. View at Publisher · View at Google Scholar · View at Scopus
  72. N. R. Bruno, “Group of boost and rotation transformations with two observer-independent scales,” Physics Letters B, vol. 547, no. 1-2, pp. 109–115, 2002. View at Google Scholar
  73. A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santander, “Classical deformations, Poisson-Lie contractions, and quantization of dual Lie bialgebras,” Journal of Mathematical Physics, vol. 36, no. 2, pp. 631–640, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  74. M. Born, “A suggestion for unifying quantum theory and relativity,” Proceedings of the Royal Society of London A, vol. 165, pp. 291–303, 1938. View at Google Scholar
  75. A. Blaut, M. Daszkiewicz, J. Kowalski-Glikman, and S. Nowak, “Phase spaces of doubly special relativity,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 582, no. 1-2, pp. 82–85, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  76. A. Y. Alekseev and A. Z. Malkin, “Symplectic structure of the moduli space of flat connection on a Riemann surface,” Communications in Mathematical Physics, vol. 169, no. 1, pp. 99–119, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  77. V. V. Fock and A. A. Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix,” ITEP-72-92, 1992.
  78. V. V. Fock and A. A. Rosly, “Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix,” American Mathematical Society Translations, vol. 191, pp. 67–86, 1999. View at Google Scholar
  79. A. Achúcarro and P. K. Townsend, “A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories,” Physics Letters B, vol. 180, no. 1-2, pp. 89–92, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  80. E. Witten, “2+1 dimensional gravity as an exactly soluble system,” Nuclear Physics B, vol. 311, no. 1, pp. 46–78, 1988. View at Google Scholar
  81. C. Meusburger and B. J. Schroers, “Quaternionic and Poisson-Lie structures in three-dimensional gravity: the cosmological constant as deformation parameter,” Journal of Mathematical Physics, vol. 49, no. 8, 083510, 27 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  82. C. Meusburger and B. J. Schroers, “Generalised Chern–Simons actions for 3d gravity and k-Poincaré symmetry,” Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, vol. 806, no. 3, pp. 462–488, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  83. G. Papageorgiou and B. Schroers, “A Chern-Simons approach to Galilean quantum gravity in dimensions,” Journal of High Energy Physics, vol. 2009, no. 11, p. 009, 2009. View at Publisher · View at Google Scholar
  84. G. Papageorgiou and B. J. Schroers, “Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra,” Journal of High Energy Physics, vol. 11, Article ID 020, 2010. View at Google Scholar
  85. A. Ballesteros, F. J. Herranz, and C. Meusburger, “Three-dimensional gravity and Drinfel'd doubles: spacetimes and symmetries from quantum deformations,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 687, no. 4-5, pp. 375–381, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  86. A. Ballesteros, F. J. Herranz, and C. Meusburger, “Drinfel'd doubles for -gravity,” Classical and Quantum Gravity, vol. 30, no. 15, Article ID 155012, p. 20, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  87. A. Ballesteros, F. J. Herranz, C. Meusburger, and P. Naranjo, “Twisted k-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes,” SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, vol. 10, article 052, p. 26, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  88. A. Ballesteros, F. J. Herranz, and C. Meusburger, “A non-commutative Drinfel'd double spacetime with cosmological constant,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 732, pp. 201–209, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  89. A. Ballesteros, C. Meusburger, and P. Naranjo, “AdS Poisson homogeneous spaces and Drinfel’d doubles,” Journal of Physics A: Mathematical and Theoretical, vol. 50, no. 39, Article ID 395202, 26 pages, 2017. View at Publisher · View at Google Scholar
  90. G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, D. V. Nanopoulos, and S. Sarkar, “Tests of quantum gravity from observations of big gamma-ray bursts,” Nature, vol. 393, pp. 763–765, 1998. View at Publisher · View at Google Scholar · View at Scopus
  91. G. Amelino-Camelia, “Doubly-special relativity: Facts, myths and some key open issues,” Symmetry, vol. 2, no. 1, pp. 230–271, 2010. View at Publisher · View at Google Scholar · View at Scopus
  92. G. Amelino-Camelia, “Quantum-spacetime phenomenology,” Living Reviews in Relativity, vol. 16, article 5, 2013. View at Publisher · View at Google Scholar
  93. D. Mattingly, “Modern tests of Lorentz invariance,” Living Reviews in Relativity, vol. 8, no. 1, Article ID 5, 84 pages, 2005. View at Publisher · View at Google Scholar · View at Scopus
  94. S. Zakrzewski, “Poisson structures on Poincaré group,” Communications in Mathematical Physics, vol. 185, no. 2, pp. 285–311, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  95. A. Borowiec, J. Lukierski, and V. N. Tolstoy, “Quantum deformations of Euclidean, Lorentz, Kleinian and quaternionic o*(4) symmetries in unified o(4;C) setting,” Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, vol. 754, pp. 176–181, 2016. View at Publisher · View at Google Scholar · View at Scopus
  96. A. Ballesteros, F. J. Herranz, and F. Musso, “On quantum deformations of (anti-)de Sitter algebras in (2+1) dimensions,” Journal of Physics: Conference Series, vol. 532, Article ID 012002, 12 pages, 2014. View at Publisher · View at Google Scholar
  97. F. J. Herranz, A. Ballesteros, and N. R. Bruno, “On anti-de Sitter and de Sitter Lie bialgebras with dimensionful deformation parameters,” Czechoslovak Journal of Physics, vol. 54, no. 11, pp. 1321–1327, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  98. A. Ballesteros, F. J. Herranz, and P. Naranjo, “Towards gravity through Drinfel'd doubles with cosmological constant,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 746, pp. 37–43, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  99. Á. Ballesteros, F. J. Herranz, F. Musso, and P. Naranjo, “The κ-(A)dS quantum algebra in (3 + 1) dimensions,” Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, vol. 766, pp. 205–211, 2017. View at Publisher · View at Google Scholar · View at Scopus
  100. F. J. Herranz, “Non-standard quantum so(3, 2) and its contractions,” Journal of Physics. A. Mathematical and General, vol. 30, no. 17, pp. 6123–6129, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  101. J. Lukierski, V. D. Lyakhovsky, and M. Mozrzymas, “k-deformations of conformal versus deformations of AdS symmetries,” Modern Physics Letters A. Particles and Fields, Gravitation, Cosmology, Nuclear Physics, vol. 18, no. 11, pp. 753–769, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  102. J. Lukierski, V. Lyakhovsky, and M. Mozrzymas, “k-deformations of Weyl and conformal symmetries,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 538, no. 3-4, pp. 375–384, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  103. N. Aizawa, F. J. Herranz, J. Negro, and M. . del Olmo, “Twisted conformal algebra so(4, 2),” Journal of Physics. A. Mathematical and General, vol. 35, no. 39, pp. 8179–8196, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  104. F. J. Herranz, “New quantum conformal algebras and discrete symmetries,” Physics Letters. B. Particle Physics, Nuclear Physics and Cosmology, vol. 543, no. 1-2, pp. 89–97, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  105. H. Steinacker, “Quantum anti-de Sitter space and sphere at roots of unity,” Advances in Theoretical and Mathematical Physics, vol. 4, no. 1, pp. 155–208, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  106. D. Jurman and H. Steinacker, “2D fuzzy anti-de Sitter space from matrix models,” Journal of High Energy Physics, vol. 2014, no. 1, article no. 100, 2014. View at Publisher · View at Google Scholar · View at Scopus
  107. M. Burić and J. Madore, “Noncommutative de Sitter and FRW spaces,” The European Physical Journal C, vol. 75, no. 10, Article ID 502, 11 pages, 2015. View at Google Scholar