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

SIGMA 6 (2010), 074, 19 pages      arXiv:1009.4762
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Snyder Space-Time: K-Loop and Lie Triple System

Florian Girelli
School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia

Received April 29, 2010, in final form September 13, 2010; Published online September 24, 2010

Different deformations of the Poincaré symmetries have been identified for various non-commutative spaces (e.g. κ-Minkowski, sl(2,R), Moyal). We present here the deformation of the Poincaré symmetries related to Snyder space-time. The notions of smooth ''K-loop'', a non-associative generalization of Abelian Lie groups, and its infinitesimal counterpart given by the Lie triple system are the key objects in the construction.

Key words: Snyder space-time; quantum group.

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  1. Snyder H., Quantized space-time, Phys. Rev. 71 (1947), 38-41.
  2. Breckenridge J.C., Elias V., Steele T.G., Massless scalar field theory in a quantised spacetime, Classical Quantum Gravity 12 (1995), 637-650, hep-th/9501108.
  3. Girelli F., Livine E.R., Field theories with homogenous momentum space, in Proceedings of 25th Max Born Symposium: The Planck Scale (Wroclaw, Poland, 2009), AIP Conf. Proc. 1196, (2009), 115-123, arXiv:0910.3107.
  4. Girelli F., Livine E.R., Scalar field theory in Snyder space-time: alternatives, arXiv:1004.0621.
  5. Battisti M.V., Meljanac S., Scalar field theory on non-commutative Snyder space-time, Phys. Rev. D 82 (2010), 024028, 9 pages, arXiv:1003.2108.
    Meljanac S., Meljanac D., Samsarov A., Stojic M., κ-deformed Snyder spacetime, Modern Phys. Lett. A 25 (2010), 579-590, arXiv:0912.5087.
    Battisti M.V., Meljanac S., Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry, Phys. Rev. D 79 (2009), 067505, 4 pages, arXiv:0812.3755.
  6. Zakrzewski S., Poisson structures on the Poincaré group, Comm. Math. Phys. 185 (1997), 285-311, q-alg/9602001.
  7. Lukierski J., Ruegg H., Nowicki A., Tolstoi V.N., q-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  8. Majid S., Ruegg H., Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B 334 (1994), 348-354, hep-th/9405107.
  9. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  10. Kowalski-Glikman J., Nowak S., Quantum κ-Poincaré algebra from de Sitter space of momenta, hep-th/0411154.
  11. Freidel L., Kowalski-Glikman J., Nowak S., Field theory on κ-Minkowski space revisited: Noether charges and breaking of Lorentz symmetry, Internat. J. Modern Phys. A 23 (2008), 2687-2718, arXiv:0706.3658.
  12. Kiechle H., Theory of K-loops, Lecture Notes in Mathematics, Vol. 1778, Springer-Verlag, Berlin, 2002.
  13. Joung E., Mourad J., Noui K., Three dimensional quantum geometry and deformed symmetry, J. Math. Phys. 50 (2009), 052503, 29 pages, arXiv:0806.4121.
  14. Sabinin L.V., Smooth quasigroups and loops, Mathematics and Its Applications, Vol. 492, Kluwer Academic Publishers, Dordrecht, 1999.
  15. Klim J., Majid S., Hopf quasigroups and the algebraic 7-sphere, J. Algebra 323 (2010), 3067-3110, arXiv:0906.5026.
    Klim J., Majid S., Bicrossproduct Hopf quasigroups, arXiv:0911.3114.
  16. Kikkawa M., Geometry of homogeneous Lie loops, Hiroshima Math. J. 5 (1975), 141-179.
  17. Ungar A.A., Thomas precession and its associated grouplike structure, Amer. J. Phys. 59 (1991), 824-834.
  18. Sabinin L.V., Sabinina L.L., Sbitneva L.V., On the notion of gyrogroup, Aequationes Math. 56 (1998), 11-17.
  19. Girelli F., Livine E.R., Special relativity as a noncommutative geometry: lessons for deformed special relativity, Phys. Rev. D 81 (2010), 085041, 17 pages, gr-qc/0407098.
  20. Mostovoy J., Perez-Izquierdo J.M., Ideals in non-associative universal enveloping algebras of Lie triple systems, math/0506179.
    Perez-Izquierdo J.M., Algebras, hyperalgebras, nonassociative bialgebras and loops, Adv. Math. 208 (2007), 834-876.
  21. Nagy G.P., The Campbell-Hausdorff series of local analytic Bruck loops, Abh. Math. Sem. Univ. Hamburg 72 (2002), 79-87.
  22. Jacobson N., General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70 (1951), 509-530.
  23. Hodge T.L., Parshall B.J., On the representation theory of Lie triple systems, Trans. Amer. Math. Soc. 354 (2002), 4359-4391.
  24. Lister W.G., A structure theory of Lie triple systems, Trans. Amer. Math. Soc. 72 (1952), 217-242.
  25. 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.
  26. Carlson C.E., Carone C.D., Zobin N., Noncommutative gauge theory without Lorentz violation, Phys. Rev. D 66 (2002), 075001, 8 pages, hep-th/0206035.
  27. Sitarz A., Noncommutative differential calculus on the κ-Minkowski space, Phys. Lett. B 349 (1995), 42-48, hep-th/9409014.
  28. Girelli F., Livine E.R., Physics of deformed special relativity, gr-qc/0412079.
    Girelli F., Livine E.R., Physics of deformed special relativity: relativity principle revisited, Braz. J. Phys. 35 (2005), 432-438, gr-qc/0412004.
    Andriot D., Lorentz precession in deformed special relativity, unpublished work.

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