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

SIGMA 10 (2014), 106, 18 pages      arXiv:1402.0397
Contribution to the Special Issue on Deformations of Space-Time and its Symmetries

$\kappa$-Deformed Phase Space, Hopf Algebroid and Twisting

Tajron Jurić a, Domagoj Kovačević b and Stjepan Meljanac a
a) Rudjer Bošković Institute, Bijenička cesta 54, HR-10000 Zagreb, Croatia
b) University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, HR-10000 Zagreb, Croatia

Received February 21, 2014, in final form November 11, 2014; Published online November 18, 2014

Hopf algebroid structures on the Weyl algebra (phase space) are presented. We define the coproduct for the Weyl generators from Leibniz rule. The codomain of the coproduct is modified in order to obtain an algebra structure. We use the dual base to construct the target map and antipode. The notion of twist is analyzed for $\kappa$-deformed phase space in Hopf algebroid setting. It is outlined how the twist in the Hopf algebroid setting reproduces the full Hopf algebra structure of $\kappa$-Poincaré algebra. Several examples of realizations are worked out in details.

Key words: noncommutative space; $\kappa$-Minkowski spacetime; Hopf algebroid; $\kappa$-Poincaré algebra; realizations; twist.

pdf (457 kb)   tex (29 kb)


  1. Amelino-Camelia G., Testable scenario for relativity with minimum-length, Phys. Lett. B 510 (2001), 255-263, hep-th/0012238.
  2. Amelino-Camelia G., Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale, Internat. J. Modern Phys. D 11 (2002), 35-59, gr-qc/0012051.
  3. Andrade F.M., Silva E.O., Effects of quantum deformation on the spin-1/2 Aharonov-Bohm problem, Phys. Lett. B 719 (2013), 467-471, arXiv:1212.1944.
  4. Andrade F.M., Silva E.O., Ferreira Jr. M.M., Rodrigues E.C., On the $\kappa$-Dirac oscillator revisited, Phys. Lett. B 731 (2014), 327-330, arXiv:1312.2973.
  5. Arzano M., Marcianò A., Fock space, quantum fields, and $\kappa$-Poincaré symmetries, Phys. Rev. D 76 (2007), 125005, 14 pages, hep-th/0701268.
  6. 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.
  7. Aschieri P., Dimitrijević M., Meyer F., Wess J., Noncommutative geometry and gravity, Classical Quantum Gravity 23 (2006), 1883-1911, hep-th/0510059.
  8. Böhm G., Hopf algebroids, arXiv:0805.3806.
  9. Böhm G., Szlachányi K., Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274 (2004), 708-750, math.QA/0302325.
  10. Bojowald M., Paily G.M., Deformed general relativity, Phys. Rev. D 87 (2013), 044044, 7 pages, arXiv:1212.4773.
  11. 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.
  12. Borowiec A., Lukierski J., Pachoł A., Twisting and $\kappa$-Poincaré, J. Phys. A: Math. Theor. 47 (2014), 405203, 12 pages, arXiv:1312.7807.
  13. 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.
  14. Borowiec A., Pachoł A., $\kappa$-Minkowski spacetimes and DSR algebras: fresh look and old problems, SIGMA 6 (2010), 086, 31 pages, arXiv:1005.4429.
  15. 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.
  16. Borowiec A., Pachoł A., Unified description for $\kappa$-deformations of orthogonal groups, Eur. Phys. J. C 74 (2014), 2812, 9 pages, arXiv:1311.4499.
  17. Bu J.-G., Kim H.-C., Yee J.H., Differential structure on $\kappa$-Minkowski spacetime realized as module of twisted Weyl algebra, Phys. Lett. B 679 (2009), 486-490, arXiv:0903.0040.
  18. 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.
  19. Daszkiewicz M., Lukierski J., Woronowicz M., Towards quantum noncommutative $\kappa$-deformed field theory, Phys. Rev. D 77 (2008), 105007, 10 pages, arXiv:0708.1561.
  20. de Boer J., Grassi P.A., van Nieuwenhuizen P., Non-commutative superspace from string theory, Phys. Lett. B 574 (2003), 98-104, hep-th/0302078.
  21. Dolan B.P., Gupta K.S., Stern A., Noncommutative BTZ black hole and discrete time, Classical Quantum Gravity 24 (2007), 1647-1655, hep-th/0611233.
  22. Doplicher S., Fredenhagen K., Roberts J.E., Spacetime quantization induced by classical gravity, Phys. Lett. B 331 (1994), 39-44.
  23. Doplicher S., Fredenhagen K., Roberts J.E., The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), 187-220, hep-th/0303037.
  24. 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.
  25. 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.
  26. Gupta K.S., Harikumar E., Jurić T., Meljanac S., Samsarov A., Effects of noncommutativity on the black hole entropy, Adv. High Energy Phys. 2014 (2014), 139172, 10 pages, arXiv:1312.5100.
  27. Gupta K.S., Meljanac S., Samsarov A., Quantum statistics and noncommutative black holes, Phys. Rev. D 85 (2012), 045029, 8 pages, arXiv:1108.0341.
  28. Harikumar E., Maxwell's equations on the $\kappa$-Minkowski spacetime and electric-magnetic duality, Europhys. Lett. 90 (2010), 21001, 6 pages, arXiv:1002.3202.
  29. Harikumar E., Jurić T., Meljanac S., Electrodynamics on $\kappa$-Minkowski space-time, Phys. Rev. D 84 (2011), 085020, 8 pages, arXiv:1107.3936.
  30. Harikumar E., Jurić T., Meljanac S., Geodesic equation in $\kappa$-Minkowski spacetime, Phys. Rev. D 86 (2012), 045002, 8 pages, arXiv:1203.1564.
  31. Harikumar E., Sivakumar M., Srinivas N., $\kappa$-deformed Dirac equation, Modern Phys. Lett. A 26 (2011), 1103-1115, arXiv:0910.5778.
  32. Jurić T., Meljanac S., Štrajn R., Differential forms and $\kappa$-Minkowski spacetime from extended twist, Eur. Phys. J. C 73 (2013), 2472, 8 pages, arXiv:1211.6612.
  33. Jurić T., Meljanac S., Štrajn R., $\kappa$-Poincaré-Hopf algebra and Hopf algebroid structure of phase space from twist, Phys. Lett. A 377 (2013), 2472-2476, arXiv:1303.0994.
  34. Jurić T., Meljanac S., Štrajn R., Twists, realizations and Hopf algebroid structure of $\kappa$-deformed phase space, Internat. J. Modern Phys. A 29 (2014), 1450022, 32 pages, arXiv:1305.3088.
  35. Jurić T., Meljanac S., Štrajn R., Universal $\kappa$-Poincaré covariant differential calculus over $\kappa$-Minkowski space, Internat. J. Modern Phys. A 29 (2014), 1450121, 14 pages, arXiv:1312.2751.
  36. Kempf A., Mangano G., Minimal length uncertainty relation and ultraviolet regularization, Phys. Rev. D 55 (1997), 7909-7920, hep-th/9612084.
  37. Kim H.-C., Lee Y., Rim C., Yee J.H., Differential structure on the $\kappa$-Minkowski spacetime from twist, Phys. Lett. B 671 (2009), 398-401, arXiv:0808.2866.
  38. Kim H.-C., Lee Y., Rim C., Yee J.H., Scalar field theory in $\kappa$-Minkowski spacetime from twist, J. Math. Phys. 50 (2009), 102304, 12 pages, arXiv:0901.0049.
  39. Kosiński P., Lukierski J., Maślanka P., Local $D=4$ field theory on $\kappa$-deformed Minkowski space, Phys. Rev. D 62 (2000), 025004, 10 pages, hep-th/9902037.
  40. 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.
  41. Kovačević D., Meljanac S., Pachoł A., Štrajn R., Generalized Poincaré algebras, Hopf algebras and $\kappa$-Minkowski spacetime, Phys. Lett. B 711 (2012), 122-127, arXiv:1202.3305.
  42. Kovačević D., Meljanac S., Samsarov A., Škoda Z., Hermitian realizations of $\kappa$-Minkowski spacetime, arXiv:1307.5772.
  43. Kowalski-Glikman J., Introduction to doubly special relativity, in Planck Scale Effects in Astrophysics and Cosmology, Lecture Notes in Phys., Vol. 669, Springer, Berlin, 2005, 131-159, hep-th/0405273.
  44. Kowalski-Glikman J., Nowak S., Doubly special relativity theories as different bases of $\kappa$-Poincaré algebra, Phys. Lett. B 539 (2002), 126-132, hep-th/0203040.
  45. 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.
  46. Kupriyanov V.G., A hydrogen atom on curved noncommutative space, J. Phys. A: Math. Theor. 46 (2013), 245303, 7 pages, arXiv:1209.6105.
  47. Kupriyanov V.G., Quantum mechanics with coordinate dependent noncommutativity, J. Math. Phys. 54 (2013), 112105, 25 pages, arXiv:1204.4823.
  48. Lu J.-H., Hopf algebroids and quantum groupoids, Internat. J. Math. 7 (1996), 47-70, q-alg/9505024.
  49. Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and $\kappa$-deformed field theory, Phys. Lett. B 293 (1992), 344-352.
  50. Lukierski J., Ruegg H., Quantum $\kappa$-Poincaré in any dimension, Phys. Lett. B 329 (1994), 189-194, hep-th/9310117.
  51. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  52. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  53. Majid S., Ruegg H., Bicrossproduct structure of $\kappa$-Poincaré group and non-commutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9404107.
  54. Meljanac S., Krešić-Jurić S., Stojić M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C 51 (2007), 229-240, hep-th/0702215.
  55. 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.
  56. 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.
  57. Meljanac S., Samsarov A., Štrajn R., $\kappa$-deformation of phase space; generalized Poincaré algebras and $R$-matrix, J. High Energy Phys. 2012 (2012), no. 8, 127, 16 pages, arXiv:1204.4324.
  58. Meljanac S., Stojić M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C 47 (2006), 531-539, hep-th/0605133.
  59. Meljanac S., Škoda Z., Lie algebra type noncommutative phase spaces are Hopf algebroids, arXiv:1409.8188.
  60. Schupp P., Solodukhin S., Exact black hole solutions in noncommutative gravity, arXiv:0906.2724.
  61. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, hep-th/9908142.
  62. Takeuchi M., Groups of algebras over $A\otimes \overline A$, J. Math. Soc. Japan 29 (1977), 459-492.
  63. Xu P., Quantum groupoids, Comm. Math. Phys. 216 (2001), 539-581, math.QA/9905192.
  64. Young C.A.S., Zegers R., Covariant particle exchange for $\kappa$-deformed theories in $1+1$ dimensions, Nuclear Phys. B 804 (2008), 342-360, arXiv:0803.2659.
  65. 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.

Previous article  Next article   Contents of Volume 10 (2014)