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Snyder Space-Time: K-Loop and Lie Triple System
Symmetry, integrability and geometry: methods and applications, Tome 6 (2010) Cet article a éte moissonné depuis la source Math-Net.Ru

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Different deformations of the Poincaré symmetries have been identified for various non-commutative spaces (e.g. $\kappa$-Minkowski, $\mathfrak{sl}(2,R)$, 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.
Keywords: Snyder space-time; quantum group. 
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     author = {Florian Girelli},
     title = {Snyder {Space-Time:} {K-Loop} and {Lie} {Triple} {System}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a73/}
}
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Florian Girelli. Snyder Space-Time: K-Loop and Lie Triple System. Symmetry, integrability and geometry: methods and applications, Tome 6 (2010). http://geodesic.mathdoc.fr/item/SIGMA_2010_6_a73/

[1] Snyder H., “Quantized space-time”, Phys. Rev., 71 (1947), 38–41 | DOI | MR | Zbl

[2] Breckenridge J. C., Elias V., Steele T. G., “Massless scalar field theory in a quantised spacetime”, Classical Quantum Gravity, 12 (1995), 637–650, arXiv: hep-th/9501108 | DOI | MR | Zbl

[3] Girelli F., Livine E. R., “Field theories with homogenous momentum space,”, Proceedings of 25th Max Born Symposium: The Planck Scale (Wroclaw, Poland, 2009), AIP Conf. Proc., 1196, 2009, 115–123, arXiv: 0910.3107 | DOI

[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 pp., arXiv: ; Meljanac S., Meljanac D., Samsarov A., Stojic M., “$\kappa$-deformed Snyder spacetime”, Modern Phys. Lett. A, 25 (2010), 579–590, arXiv: ; Battisti M. V., Meljanac S., “Modification of Heisenberg uncertainty relations in non-commutative Snyder space-time geometry”, Phys. Rev. D, 79 (2009), 067505, 4 pp., arXiv: 1003.21080912.50870812.3755 | DOI | DOI | MR | Zbl | DOI | MR

[6] Zakrzewski S., “Poisson structures on the Poincaré group”, Comm. Math. Phys., 185 (1997), 285–311, arXiv: q-alg/9602001 | DOI | MR | Zbl

[7] Lukierski J., Ruegg H., Nowicki A., Tolstoi V. N., “$q$-deformation of Poincaré algebra”, Phys. Lett. B, 264 (1991), 331–338 | DOI | MR

[8] Majid S., Ruegg H., “Bicrossproduct structure of $\kappa$-Poincaré group and noncommutative geometry”, Phys. Lett. B, 334 (1994), 348–354, arXiv: hep-th/9405107 | DOI | MR | Zbl

[9] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995 | DOI | MR | Zbl

[10] Kowalski-Glikman J., Nowak S., Quantum $\kappa$-Poincaré algebra from de Sitter space of momenta, arXiv: hep-th/0411154

[11] 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 | DOI | MR | Zbl

[12] Kiechle H., Theory of $K$-loops, Lecture Notes in Mathematics, 1778, Springer-Verlag, Berlin, 2002 | DOI | MR | Zbl

[13] Joung E., Mourad J., Noui K., “Three dimensional quantum geometry and deformed symmetry”, J. Math. Phys., 50 (2009), 052503, 29 pp., arXiv: 0806.4121 | DOI | MR | Zbl

[14] Sabinin L. V., Smooth quasigroups and loops, Mathematics and Its Applications, 492, Kluwer Academic Publishers, Dordrecht, 1999 | MR | Zbl

[15] Klim J., Majid S., “Hopf quasigroups and the algebraic 7-sphere”, J. Algebra, 323 (2010), 3067–3110, arXiv: ; Klim J., Majid S., Bicrossproduct Hopf quasigroups, arXiv: 0906.50260911.3114 | DOI | Zbl

[16] Kikkawa M., “Geometry of homogeneous Lie loops”, Hiroshima Math. J., 5 (1975), 141–179 | MR | Zbl

[17] Ungar A. A., “Thomas precession and its associated grouplike structure”, Amer. J. Phys., 59 (1991), 824–834 | DOI | MR

[18] Sabinin L. V., Sabinina L. L., Sbitneva L. V., “On the notion of gyrogroup”, Aequationes Math., 56 (1998), 11–17 | DOI | MR | Zbl

[19] Girelli F., Livine E. R., “Special relativity as a noncommutative geometry: lessons for deformed special relativity”, Phys. Rev. D, 81 (2010), 085041, 17 pp., arXiv: gr-qc/0407098 | DOI | MR

[20] Mostovoy J., Perez-Izquierdo J. M., Ideals in non-associative universal enveloping algebras of Lie triple systems, arXiv: ; Perez-Izquierdo J. M., “Algebras, hyperalgebras, nonassociative bialgebras and loops”, Adv. Math., 208 (2007), 834–876 math/0506179 | DOI | MR | Zbl

[21] Nagy G. P., “The Campbell–Hausdorff series of local analytic Bruck loops”, Abh. Math. Sem. Univ. Hamburg, 72 (2002), 79–87 | DOI | MR | Zbl

[22] Jacobson N., “General representation theory of Jordan algebras”, Trans. Amer. Math. Soc., 70 (1951), 509–530 | DOI | MR | Zbl

[23] Hodge T. L., Parshall B. J., “On the representation theory of Lie triple systems”, Trans. Amer. Math. Soc., 354 (2002), 4359–4391 | DOI | MR | Zbl

[24] Lister W. G., “A structure theory of Lie triple systems”, Trans. Amer. Math. Soc., 72 (1952), 217–242 | DOI | MR | Zbl

[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, arXiv: hep-th/0303037 | DOI | MR | Zbl

[26] Carlson C. E., Carone C. D., Zobin N., “Noncommutative gauge theory without Lorentz violation”, Phys. Rev. D, 66 (2002), 075001, 8 pp., arXiv: hep-th/0206035 | DOI

[27] Sitarz A., “Noncommutative differential calculus on the $\kappa$-Minkowski space”, Phys. Lett. B, 349 (1995), 42–48, arXiv: hep-th/9409014 | DOI | MR

[28] Girelli F., Livine E. R., Physics of deformed special relativity, arXiv: ; Girelli F., Livine E. R., “Physics of deformed special relativity: relativity principle revisited”, Braz. J. Phys., 35 (2005), 432–438, arXiv: ; Andriot D., Lorentz precession in deformed special relativity, unpublished work gr-qc/0412079gr-qc/0412004 | DOI | MR