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

SIGMA 10 (2014), 107, 24 pages      arXiv:1404.2916      https://doi.org/10.3842/SIGMA.2014.107
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

$\kappa$-Deformations and Extended $\kappa$-Minkowski Spacetimes

Andrzej Borowiec a and Anna Pachoł b, c
a) Institute for Theoretical Physics, pl. M. Borna 9, 50-204 Wrocław, Poland
b) Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland
c) Capstone Institute for Theoretical Research, Reykjavik, Iceland

Received April 11, 2014, in final form November 11, 2014; Published online November 22, 2014

Abstract
We extend our previous study of Hopf-algebraic $\kappa$-deformations of all inhomogeneous orthogonal Lie algebras ${\rm iso}(g)$ as written in a tensorial and unified form. Such deformations are determined by a vector $\tau$ which for Lorentzian signature can be taken time-, light- or space-like. We focus on some mathematical aspects related to this subject. Firstly, we describe real forms with connection to the metric's signatures and their compatibility with the reality condition for the corresponding $\kappa$-Minkowski (Hopf) module algebras. Secondly, $h$-adic vs $q$-analog (polynomial) versions of deformed algebras including specialization of the formal deformation parameter $\kappa$ to some numerical value are considered. In the latter the general covariance is lost and one deals with an orthogonal decomposition. The last topic treated in this paper concerns twisted extensions of $\kappa$-deformations as well as the description of resulting noncommutative spacetime algebras in terms of solvable Lie algebras. We found that if the type of the algebra does not depend on deformation parameters then specialization is possible.

Key words: quantum deformations; quantum groups; quantum spaces; reality condition for Hopf module algebras; $q$-analog and specialization versions; $\kappa$-Minkowski spacetime; extended $\kappa$-deformations; twist-deformations; classification of solvable Lie algebras.

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References

  1. Amelino-Camelia G., Mandanici G., Procaccini A., Kowalski-Glikman J., Phenomenology of doubly special relativity, Internat. J. Modern Phys. A 20 (2005), 6007-6037, gr-qc/0312124.
  2. Amelino-Camelia G., Smolin L., Prospects for constraining quantum gravity dispersion with near term observations, Phys. Rev. D 80 (2009), 084017, 14 pages, arXiv:0906.3731.
  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. Aschieri P., Blohmann C., Dimitrijević M., Meyer F., Schupp P., Wess J., A gravity theory on noncommutative spaces, Classical Quantum Gravity 22 (2005), 3511-3532, hep-th/0504183.
  5. Aschieri P., Castellani L., Noncommutative gravity solutions, J. Geom. Phys. 60 (2010), 375-393, arXiv:0906.2774.
  6. Aschieri P., Dimitrijević M., Meyer F., Schraml S., Wess J., Twisted gauge theories, Lett. Math. Phys. 78 (2006), 61-71, hep-th/0603024.
  7. Aschieri P., Dimitrijević M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity 23 (2006), 1883-1911, hep-th/0510059.
  8. Aschieri P., Schenkel A., Noncommutative connections on bimodules and Drinfeld twist deformation, Adv. Theor. Math. Phys. 18 (2014), 513-612, arXiv:1210.0241.
  9. Ballesteros A., 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.
  10. 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.
  11. Ballesteros A., 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.
  12. Ballesteros Á., Herranz F.J., Meusburger C., Naranjo P., Twisted $(2+1)$ $\kappa$-AdS algebra, Drinfel'd doubles and non-commutative spacetimes, SIGMA 10 (2014), 052, 26 pages, arXiv:1403.4773.
  13. Ballesteros A., Herranz F.J., Pereña C.M., Null-plane quantum universal $R$-matrix, Phys. Lett. B 391 (1997), 71-77, q-alg/9607009.
  14. Ballesteros A., Herranz F.J., Bruno N.R., Quantum (anti)de Sitter algebras and generalizations of the kappa-Minkowski space, in Proceedings of 11th International Conference on Symmetry Methods in Physics (June 21-24, 2004, Prague), Joint Institute for Nuclear Research, Dubna, 2004, 1-20, hep-th/0409295.
  15. Bonneau P., Flato M., Gerstenhaber M., Pinczon G., The hidden group structure of quantum groups: strong duality, rigidity and preferred deformations, Comm. Math. Phys. 161 (1994), 125-156.
  16. Bonneau P., Flato M., Pinczon G., A natural and rigid model of quantum groups, Lett. Math. Phys. 25 (1992), 75-84.
  17. Borowiec A., Gupta K.S., Meljanac S., Pachoł A., Constraints on the quantum gravity scale from $\kappa$-Minkowski spacetime, Europhys. Lett. 92 (2010), 20006, 6 pages, arXiv:0912.3299.
  18. Borowiec A., Lukierski J., Pachoł A., Twisting and $\kappa$-Poincaré, J. Phys. A: Math. Theor. 47 (2014), 405203, 12 pages, arXiv:1312.7807.
  19. Borowiec A., Lukierski J., Tolstoy V.N., Jordanian quantum deformations of $D=4$ anti-de Sitter and Poincaré superalgebras, Eur. Phys. J. C 44 (2005), 139-145, hep-th/0412131.
  20. Borowiec A., Lukierski J., Tolstoy V.N., On twist quantizations of $D=4$ Lorentz and Poincaré algebras, Czechoslovak J. Phys. 55 (2005), 1351-1356, hep-th/0510154.
  21. Borowiec A., Lukierski J., Tolstoy V.N., Jordanian twist quantization of $D=4$ Lorentz and Poincaré algebras and $D=3$ contraction limit, Eur. Phys. J. C 48 (2006), 633-639, hep-th/0604146.
  22. Borowiec A., Lukierski J., Tolstoy V.N., Quantum deformations of $D=4$ Lorentz algebra revisited: twistings of $q$-deformation, Eur. Phys. J. C 57 (2008), 601-611, arXiv:0804.3305.
  23. Borowiec A., Pachoł A., $\kappa$-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  24. Borowiec A., Pachoł A., $\kappa$-Minkowski spacetimes and DSR algebras: fresh look and old problems, SIGMA 6 (2010), 086, 31 pages, arXiv:1005.4429.
  25. Borowiec A., Pachoł A., The classical basis for the $\kappa$-Poincaré Hopf algebra and doubly special relativity theories, J. Phys. A: Math. Theor. 43 (2010), 045203, 10 pages, arXiv:0903.5251.
  26. Borowiec A., Pachoł A., Unified description for $\kappa$-deformations of orthogonal groups, Eur. Phys. J. C 74 (2014), 2812, 9 pages, arXiv:1311.4499.
  27. 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.
  28. Bu J.-G., Kim H.-C., Lee Y., Vac C.H., Yee J.H., $\kappa$-deformed spacetime from twist, Phys. Lett. B 665 (2008), 95-99, hep-th/0611175.
  29. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  30. D'Andrea F., Spectral geometry of $\kappa$-Minkowski space, J. Math. Phys. 47 (2006), 062105, 19 pages, hep-th/0503012.
  31. Daszkiewicz M., Generalized twist deformations of Poincaré and Galilei Hopf algebras, Rep. Math. Phys. 63 (2009), 263-277, arXiv:0812.1613.
  32. Daszkiewicz M., Imiłkowska K., Kowalski-Glikman J., Nowak S., Scalar field theory on $\kappa$-Minkowski space-time and doubly special relativity, Internat. J. Modern Phys. A 20 (2005), 4925-4940, hep-th/0410058.
  33. Daszkiewicz M., Lukierski J., Woronowicz M., $\kappa$-deformed statistics and classical four-momentum addition law, Modern Phys. Lett. A 23 (2008), 653-665, hep-th/0703200.
  34. Daszkiewicz M., Lukierski J., Woronowicz M., Towards quantum noncommutative $\kappa$-deformed field theory, Phys. Rev. D 77 (2008), 105007, 10 pages, arXiv:0708.1561.
  35. Daszkiewicz M., Lukierski J., Woronowicz M., $\kappa$-deformed oscillators, the choice of star product and free $\kappa$-deformed quantum fields, J. Phys. A: Math. Theor. 42 (2009), 355201, 18 pages, arXiv:0807.1992.
  36. de Graaf W.A., Classification of solvable Lie algebras, Experiment. Math. 14 (2005), 15-25, math.RA/0404071.
  37. Dimitrijević M., Jonke L., Möller L., Tsouchnika E., Wess J., Wohlgenannt M., Deformed field theory on $\kappa$-spacetime, Eur. Phys. J. C 31 (2003), 129-138, hep-th/0307149.
  38. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  39. Drinfel'd V.G., Quasi-Hopf algebras, Leningrad Math. J. 1 (1989), 1419-1457.
  40. Durhuus B., Sitarz A., Star product realizations of $\kappa$-Minkowski space, J. Noncommut. Geom. 7 (2013), 605-645, arXiv:1104.0206.
  41. Freidel L., Kowalski-Glikman J., Nowak S., Field theory on $\kappa$-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry, Internat. J. Modern Phys. A 23 (2008), 2687-2718, arXiv:0706.3658.
  42. Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Twisted statistics in $\kappa$-Minkowski spacetime, Phys. Rev. D 77 (2008), 105010, 6 pages, arXiv:0802.1576.
  43. Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Deformed osciallator algebras and QFT in the $\kappa$-Minkowski spacetime, Phys. Rev. D 80 (2009), 025014, 11 pages, arXiv:0903.2355.
  44. Gupta K.S., Meljanac S., Samsarov A., Quantum statistics and noncommutative black holes, Phys. Rev. D 85 (2012), 045029, 8 pages, arXiv:1108.0341.
  45. Harikumar E., Jurić T., Meljanac S., Geodesic equation in $\kappa$-Minkowski spacetime, Phys. Rev. D 86 (2012), 045002, 8 pages, arXiv:1203.1564.
  46. Hossenfelder S., Minimal length scale scenarios for quantum gravity, Living Rev. Relativity 16 (2013), 2, 90 pages, arXiv:1203.6191.
  47. Iochum B., Masson T., Schücker T., Sitarz A., Compact $\kappa$-deformation and spectral triples, Rep. Math. Phys. 68 (2011), 37-64, arXiv:1004.4190.
  48. Iochum B., Masson T., Schücker T., Sitarz A., $\kappa$-deformation and spectral triples, Acta Phys. Polon. B Proc. Suppl. 4 (2011), 305-324, arXiv:1107.3449.
  49. Jurčo B., Möller L., Schraml S., Schupp P., Wess J., Construction of non-abelian gauge theories on noncommutative spaces, Eur. Phys. J. C 21 (2001), 383-388, hep-th/0104153.
  50. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  51. Kosiński P., Maślanka P., The $\kappa$-Weyl group and its algebra, in From Field Theory to Quantum Groups, World Sci. Publ., River Edge, NJ, 1996, 41-51, q-alg/9512018.
  52. Kovačević D., Meljanac S., Kappa-Minkowski spacetime, kappa-Poincaré Hopf algebra and realizations, J. Phys. A: Math. Theor. 45 (2012), 135208, 24 pages, arXiv:1110.0944.
  53. Kowalski-Glikman J., Observer-independent quantum of mass, Phys. Lett. A 286 (2001), 391-394, hep-th/0102098.
  54. Kulish P.P., Lyakhovsky V.D., Mudrov A.I., Extended Jordanian twists for Lie algebras, J. Math. Phys. 40 (1999), 4569-4586, math.QA/9806014.
  55. Lukierski J., Lyakhovsky V.D., Two-parameter extensions of the $\kappa$-Poincaré quantum deformation, in Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math., Vol. 391, Amer. Math. Soc., Providence, RI, 2005, 281-288, hep-th/0406155.
  56. Lukierski J., Lyakhovsky V.D., Mozrzymas M., $\kappa$-deformations of $D=4$ Weyl and conformal symmetries, Phys. Lett. B 538 (2002), 375-384, hep-th/0203182.
  57. Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and $\kappa$-deformed field theory, Phys. Lett. B 293 (1992), 344-352.
  58. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  59. 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.
  60. Lukierski J., Woronowicz M., New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries, Phys. Lett. B 633 (2006), 116-124, hep-th/0508083.
  61. Lyakhovsky V.D., Twist deformations of $\kappa$-Poincaré algebra, Rep. Math. Phys. 61 (2008), 213-220.
  62. Madore J., Schraml S., Schupp P., Wess J., Gauge theory on noncommutative spaces, Eur. Phys. J. C 16 (2000), 161-167, hep-th/0001203.
  63. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  64. Majid S., Ruegg H., Bicrossproduct structure of $\kappa$-Poincaré group and non-commutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9405107.
  65. Matschull H.-J., Welling M., Quantum mechanics of a point particle in $(2+1)$-dimensional gravity, Classical Quantum Gravity 15 (1998), 2981-3030, gr-qc/9708054.
  66. Meljanac S., Krešić-Jurić S., Stojić M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C 51 (2007), 229-240, hep-th/0702215.
  67. Meljanac S., Samsarov A., Scalar field theory on $\kappa$-Minkowski space-time and translation and Lorentz invariance, Internat. J. Modern Phys. A 26 (2011), 1439-1468, arXiv:1007.3943.
  68. Meljanac S., Samsarov A., Stojić M., Gupta K.S., $\kappa$-Minkowski spacetime and the star product realizations, Eur. Phys. J. C 53 (2008), 295-309, arXiv:0705.2471.
  69. Meljanac S., Samsarov A., Trampetić J., Wohlgenannt M., Scalar field propagation in the $\phi^4$ kappa-Minkowski model, J. High Energy Phys. 2011 (2011), no. 12, 010, 23 pages, arXiv:1111.5553.
  70. Mercati F., Sitarz A., $\kappa$-Minkowski differential calculi and star product, PoS Proc. Sci. (2010), PoS(CNCFG2010), 030, 11 pages, arXiv:1105.1599.
  71. Meusburger C., Schroers B.J., Generalised Chern-Simons actions for 3d gravity and $\kappa$-Poincaré symmetry, Nuclear Phys. B 806 (2009), 462-488, arXiv:0805.3318.
  72. Mubarakzjanov G.M., On solvable Lie algebras, Izv. Vys. Ucheb. Zaved. Matematika (1963), no. 1(32), 114-123.
  73. Mudrov A.I., Twisting cocycle for null-plane quantized Poincaré algebra, J. Phys. A: Math. Gen. 31(1998), 6219-6224, q-alg/9711001.
  74. Ohl T., Schenkel A., Cosmological and black hole spacetimes in twisted noncommutative gravity, J. High Energy Phys. 2009 (2009), no. 10, 052, 12 pages, arXiv:0906.2730.
  75. Oriti D., Emergent non-commutative matter fields from group field theory models of quantum spacetime, J. Phys. Conf. Ser. 174 (2009), 012047, 14 pages, arXiv:0903.3970.
  76. Patera J., Sharp R.T., Winternitz P., Zassenhaus H., Invariants of real low dimension Lie algebras, J. Math. Phys. 17 (1976), 986-994.
  77. Podleś P., Woronowicz S.L., On the classification of quantum Poincaré groups, Comm. Math. Phys. 178 (1996), 61-82, hep-th/9412059.
  78. Podleś P., Woronowicz S.L., On the structure of inhomogeneous quantum groups, Comm. Math. Phys. 185 (1997), 325-358, hep-th/9412058.
  79. Popovych R.O., Boyko V.M., Nesterenko M.O., Lutfullin M.W., Realizations of real low-dimensional Lie algebras, J. Phys. A: Math. Gen. 36 (2003), 7337-7360, math-ph/0301029.
  80. Schenkel A., Uhlemann C.F., Field theory on curved noncommutative spacetimes, SIGMA 6 (2010), 061, 19 pages, arXiv:1003.3190.
  81. Sitarz A., Twists and spectral triples for isospectral deformations, Lett. Math. Phys. 58 (2001), 69-79, math.QA/0102074.
  82. Stachura P., Towards a topological (dual of) quantum $\kappa$-Poincaré group, Rep. Math. Phys. 57 (2006), 233-256, hep-th/0505093.
  83. Tolstoy V.N., Quantum deformations of relativistic symmetries, Invited talk at the XXII Max Born Symposium ''Quantum, Super and Twistors'' (September 27-29, 2006, Wroclaw, Poland), in honour of Jerzy Lukierski, arXiv:0704.0081.
  84. Tolstoy V.N., Twisted quantum deformations of Lorentz and Poincaré algebras, Invited talk at the VII International Workshop ''Lie Theory and its Applications in Physics'' (June 18-24, 2007, Varna, Bulgaria), arXiv:0712.3962.
  85. Young C.A.S., Zegers R., Covariant particle statistics and intertwiners of the $\kappa$-deformed Poincaré algebra, Nuclear Phys. B 797 (2008), 537-549, arXiv:0711.2206.
  86. Young C.A.S., Zegers R., Deformation quasi-Hopf algebras of non-semisimple type from cochain twists, Comm. Math. Phys. 298 (2010), 585-611, arXiv:0812.3257.
  87. Zakrzewski S., Quantum Poincaré group related to the $\kappa$-Poincaré algebra, J. Phys. A: Math. Gen. 27 (1994), 2075-2082.
  88. Zakrzewski S., Poisson structures on Poincaré group, Comm. Math. Phys. 185 (1997), 285-311, q-alg/9602001.

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