$\kappa$-Minkowski Spacetimes and DSR Algebras: Fresh Look and Old Problems

Andrzej Borowiec ; Anna Pachol

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

Some classes of Deformed Special Relativity (DSR) theories are reconsidered within the Hopf algebraic formulation. For this purpose we shall explore a minimal framework of deformed Weyl–Heisenberg algebras provided by a smash product construction of DSR algebra. It is proved that this DSR algebra, which uniquely unifies $\kappa$-Minkowski spacetime coordinates with Poincaré generators, can be obtained by nonlinear change of generators from undeformed one. Its various realizations in terms of the standard (undeformed) Weyl–Heisenberg algebra opens the way for quantum mechanical interpretation of DSR theories in terms of relativistic (Stückelberg version) Quantum Mechanics. On this basis we review some recent results concerning twist realization of $\kappa$-Minkowski spacetime described as a quantum covariant algebra determining a deformation quantization of the corresponding linear Poisson structure. Formal and conceptual issues concerning quantum $\kappa$-Poincaré and $\kappa$-Minkowski algebras as well as DSR theories are discussed. Particularly, the so-called "$q$-analog" version of DSR algebra is introduced. Is deformed special relativity quantization of doubly special relativity remains an open question. Finally, possible physical applications of DSR algebra to description of some aspects of Planck scale physics are shortly recalled.

Détail

Keywords: quantum deformations; quantum groups; Hopf module algebras; covariant quantum spaces; crossed product algebra; twist quantization, quantum Weyl algebra, $\kappa$-Minkowski spacetime; deformed phase space; quantum gravity scale; deformed dispersion relations; time delay.

Bibliographie

[1] Zakrzewski S., “Quantum Poincaré group related to the $\kappa$-Poincaré algebra”, J. Phys. A: Math. Gen., 27 (1994), 2075–2082 | DOI | MR | Zbl

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

[3] Lukierski J., Ruegg H., Zakrzewski W. J., “Classical and quantum mechanics of free $k$-relativistic systems”, Ann. Physics, 243 (1995), 90–116, arXiv: hep-th/9312153 | DOI | MR | Zbl

[4] Lukierski J., Nowicki A., Ruegg H., Tolstoy V. N., “$q$-deformation of Poincaré algebra”, Phys. Lett. B, 264 (1991), 331–338 ; Lukierski J., Ruegg H., “Quantum $\kappa$-Poincaré in any dimension”, Phys. Lett. B, 329 (1994), 189–194, arXiv: hep-th/9310117 | DOI | MR | DOI | MR

[5] Maślanka P., “The induced representations of the $\kappa$-Poincaré group. The massive case”, J. Math. Phys., 35 (1994), 5047–5056 ; Kosiński P., Maślanka P., “The $\kappa$-Weyl group and its algebra”, From Field Theory to Quantum Groups, eds. B. Jancewicz and J. Sobczyk, World Scientific, 1996, 41–49, arXiv: ; Kosiński P., Maślanka P., “On the definition of velocity in doubly special relativity theories”, Phys. Rev. D, 68 (2003), 067702, 4 pp., arXiv: q-alg/9512018hep-th/0211057 | DOI | MR | Zbl | MR | DOI | MR

[6] Kosiński P., Lukierski J., Maślanka P., Sobczyk J., “The classical basis for $\kappa$-deformed Poincaré algebra and superalgebra”, Modern Phys. Lett. A, 10 (1995), 2599–2606, arXiv: hep-th/9412114 | DOI | MR | Zbl

[7] Lukierski J., Nowicki A., “Heisenberg double description of $\kappa$-Poincaré algebra and $\kappa$-deformed phase space”, Quantum Group Symposium of Group 21, Proceedings of the XXI International Colloquium on Group Theoretical Methods in Physics, eds. V. K. Dobrev and H. D. Doebner, Heron Press, Sofia, 1997, 186–192, arXiv: ; Amelino-Camelia G., Lukierski J., Nowicki A., “$\kappa$-deformed covariant phase space and quantum-gravity uncertainty relations”, Phys. Atomic Nuclei, 61 (1998), 1811–1815, arXiv: q-alg/9702003hep-th/9706031 | MR

[8] Nowicki A., $\kappa$-deformed phase space and uncertainty relations, arXiv: math.QA/9803064

[9] Kowalski-Glikman J., Nowak S., “Doubly special relativity theories as different bases of $\kappa$-Poincaré algebra”, Phys. Lett. B, 539 (2002), 126–132, arXiv: ; 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: hep-th/02030400706.3658 | DOI | MR | Zbl | DOI | MR | Zbl

[10] Lukierski J., “$\kappa$-deformations of relativistic symmetries: some recent developments”, Quantum Group Symposium of Group 21, Proceedings of the XXI International Colloquium on Group Theoretical Methods in Physics, eds. V. K. Dobrev and H. D. Doebner, Heron Press, Sofia, 1997, 173–180; Lukierski J., Nowicki A., “Doubly Special Relativity versus $\kappa$-deformation of relativistic kinematics”, Internat. J. Modern Phys. A, 18 (2003), 7–18, arXiv: hep-th/0203065 | DOI | MR | Zbl

[11] Ballesteros A., Bruno N. R., Herranz F. J., “A non-commutative Minkowskian spacetime from a quantum AdS algebra”, Phys. Lett. B, 574 (2003), 276–282, arXiv: ; Ballesteros A., Herranz F. J., Bruno N. R., “Quantum (anti)de Sitter algebras and generalizations of the $\kappa$-Minkowski space”, Symmetry Methods in Physics, eds. C. Burdik, O. Navratil and S. Posta, Joint Institute for Nuclear Research, Dubna, Russia, 2004, 1–20, arXiv: ; Ballesteros A., Bruno N. R., Herranz F. J., “A new ‘doubly special relativity’ theory from a quantum Weyl–Poincaré algebra”, J. Phys. A: Math. Gen., 36 (2003), 10493–10503, arXiv: ; Herranz F. J., “New quantum conformal algebras and discrete symmetries”, Phys. Lett. B, 543 (2002), 89–97, arXiv: hep-th/0306089hep-th/0409295hep-th/0305033hep-ph/0205190 | DOI | MR | Zbl | MR | DOI | MR | Zbl | DOI | MR | Zbl

[12] Meljanac S., Stojić M., “New realizations of Lie algebra kappa-deformed Euclidean space”, Eur. Phys. J. C, 47 (2006), 531–539, arXiv: hep-th/0605133 | DOI | MR | Zbl

[13] 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, arXiv: hep-th/0611175 | DOI | MR

[14] Meljanac S., Krešić-Jurić S., Stojić M., “Covariant realizations of kappa-deformed space”, Eur. Phys. J. C, 51 (2007), 229–240, arXiv: hep-th/0702215 | DOI | MR | Zbl

[15] 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 | DOI | MR | Zbl

[16] Borowiec A., Pachoł A., “$\kappa$-Minkowski spacetime as the result of Jordanian twist deformation”, Phys. Rev. D, 79 (2009), 045012, 11 pp., arXiv: 0812.0576 | DOI

[17] 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 pp., arXiv: 0903.5251 | DOI | MR | Zbl

[18] Borowiec A., Gupta K. S., Meljanac S., Pachoł A., “Constraints on the quantum gravity scale from $\kappa$-Minkowski spacetime”, Eur. Phys. Lett. (to appear) , arXiv: 0912.3299

[19] Dabrowski L., Piacitelli G., Poincaré covariant $\kappa$-Minkowski spacetime, arXiv: 1006.5658

[20] Doplicher S., Fredenhagen K., Roberts J. E., “Spacetime quantization induced by classical gravity”, Phys. Lett. B, 331 (1994), 39–44 ; 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 | DOI | MR | Zbl

[21] Oeckl R., “Untwisting noncommutative $\mathbf R^d$ and the equivalence of quantum field theories”, Nuclear Phys. B, 581 (2000), 559–574, arXiv: hep-th/0003018 | DOI | MR | Zbl

[22] 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, arXiv: hep-th/0504183 | DOI | MR | Zbl

[23] Aschieri P., Jurčo B., Schupp P., Wess J., “Non-commutative GUTs, Standard Model and $C$, $P$, $T$”, Nuclear Phys B, 651 (2003), 45–70, arXiv: hep-th/0205214 | DOI | MR | Zbl

[24] Madore J., Schraml S., Schupp P., Wess J., “Gauge theory on noncommutative spaces”, Eur. Phys. J. C, 16 (2000), 161–167, arXiv: hep-th/0001203 | DOI | MR

[25] Jurčo B., Schraml S., Schupp P., Wess J., “Enveloping algebra-valued gauge transformations for non-abelian gauge groups on non-commutative spaces”, Eur. Phys. J. C, 17 (2000), 521–526, arXiv: hep-th/0006246 | DOI | MR | Zbl

[26] 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, arXiv: hep-th/0104153 | DOI | MR | Zbl

[27] Aschieri P., Dimitrijević M., Meyer F., Schraml S., Wess J., “Twisted gauge theories”, Lett. Math. Phys., 78 (2006), 61–71, arXiv: hep-th/0603024 | DOI | MR | Zbl

[28] Aschieri P., Dimitrijević M., Meyer F., Wess J., “Noncommutative geometry and gravity”, Classical Quantum Gravity, 23 (2006), 1883–1911, arXiv: hep-th/0510059 | DOI | MR | Zbl

[29] Szabo R. J., “Symmetry, gravity and noncommutativity”, Classical Quantum Gravity, 23 (2006), R199–R242, arXiv: hep-th/0606233 | DOI | MR | Zbl

[30] Chaichian M., Kulish P. P., Nishijima K., Tureanu A., “On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT”, Phys. Lett. B, 604 (2004), 98–102, arXiv: hep-th/0408069 | DOI | MR

[31] Chaichian M., Prešnajder P., Tureanu A., “New concept of relativistic invariance in noncommutative space-time: twisted Poincaré symmetry and its implications”, Phys. Rev. Lett., 94 (2005), 151602, 4 pp., arXiv: hep-th/0409096 | DOI

[32] 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, arXiv: hep-th/0307149 | DOI | MR | Zbl

[33] Freidel L., Kowalski-Glikman J., Nowak S., “From noncommutative $\kappa$-Minkowski to Minkowski space-time”, Phys. Lett. B, 648 (2007), 70–75, arXiv: hep-th/0612170 | DOI | MR

[34] 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 pp., arXiv: ; Meljanac S., Krešić-Jurić S., “Generalized kappa-deformed spaces, star products, and their realizations”, J. Phys. A: Math. Theor., 41 (2008), 235203, 24 pp., arXiv: ; Krešić-Jurić S., Meljanac S., Stojić M., “Covariant realizations of kappa-deformed space”, Eur. Phys. J. C, 51 (2007), 229–240, arXiv: 0802.15760804.3072hep-th/0702215 | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[35] Dimitrijević M., Jonke L., Möller L., Wess J., “Gauge theories on the $\kappa$-Minkowski spacetime”, Eur. Phys. J. C, 36 (2004), 117–126, arXiv: hep-th/0310116 | DOI | MR

[36] Amelino-Camelia G., “Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale”, Internat. J. Modern Phys. D, 11 (2002), 35–59, arXiv: ; Amelino-Camelia G., “Testable scenario for relativity with minimum length”, Phys. Lett. B, 510 (2001), 255–263, arXiv: ; Amelino-Camelia G., Gubitosi G., Marciano A., Martinetti P., Mercati F., “A no-pure-boost uncertainty principle from spacetime noncommutativity”, Phys. Lett. B, 671 (2008), 298–302, arXiv: gr-qc/0012051hep-th/00122381004.4190 | DOI | MR | Zbl | DOI | Zbl | DOI | MR

[37] Bruno B., Amelino-Camelia G., Kowalski-Glikman J., “Deformed boost transformations that saturate at the Planck scale”, Phys. Lett. B, 522 (2001), 133–138, arXiv: hep-th/0107039 | DOI | Zbl

[38] Magueijo J., Smolin L., “Lorentz invariance with an invariant energy scale”, Phys. Rev. Lett., 88 (2002), 190403, 4 pp., arXiv: ; Magueijo J., Smolin L., “Generalized Lorentz invariance with an invariant energy scale”, Phys. Rev. D, 67 (2003), 044017, 12 pp., arXiv: hep-th/0112090gr-qc/0207085 | DOI | DOI | MR

[39] Girelli F., Livine E. R., “Physics of deformed special relativity: relativity principle revisited”, Braz. J. Phys., 35 (2005), 432–438, arXiv: ; Girelli F., Livine E. R., Physics of deformed special relativity: relativity principle revisited, arXiv: gr-qc/0412079gr-qc/0412004 | DOI

[40] Ahluwalia-Khalilova D. V., “A freely falling frame at the interface of gravitational and quantum realms”, Classical Quantum Gravity, 22 (2005), 1433–1450, arXiv: ; Kostelecky A., Mewes M., “Electrodynamics with Lorentz-violating operators of arbitrary dimension”, Phys. Rev. D, 80 (2009), 015020, 59 pp., arXiv: hep-th/05031410905.0031 | DOI | MR | Zbl | DOI | MR

[41] Amelino-Camelia G., Smolin L., “Prospects for constraining quantum gravity dispersion with near term observations”, Phys. Rev. D, 80 (2009), 084017, 14 pp., arXiv: 0906.3731 | DOI

[42] Liberati S., Sonego S., Visser M., “Interpreting doubly special relativity as a modified theory of measurement”, Phys. Rev. D, 71 (2005), 045001, 9 pp., arXiv: gr-qc/0410113 | DOI | MR

[43] Amelino-Camelia G., “Doubly-special relativity: facts, myths and some key open issues”, Symmetry, 2 (2010), 230–271, arXiv: 1003.3942 | DOI

[44] Hossenfelder S., The box-problem in deformed special relativity, arXiv: ; Hossenfelder S., “Bounds on an energy-dependent and observer-independent speed of light from violations of locality”, Phys. Rev. Lett., 104 (2010), 140402, 4 pp., arXiv: ; Hossenfelder S., Comments on nonlocality in deformed special relativity, in reply to arXiv:1004.0664 by Lee Smolin and arXiv:1004.0575 by Jacob et al., arXiv: ; Hossenfelder S., Reply to arXiv:1006.2126 by Giovanni Amelino-Camelia et al., arXiv: ; Smolin L., Classical paradoxes of locality and their possible quantum resolutions in deformed special relativity, arXiv: 0912.00901004.04181005.05351006.45871004.0664 | DOI

[45] Jacob U., Mercati F., Amelino-Camelia G., Piran T., Modifications to Lorentz invariant dispersion in relatively boosted frames, arXiv: 1004.0575

[46] Drinfeld V., “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, 1986), Amer. Math. Soc., Providence, RI, 1987, 798–820 ; Drinfeld V., “Hopf algebras and the quantum Yang–Baxter equations”, Sov. Math. Dokl., 32 (1985), 254–258 | MR | MR

[47] Jimbo M., “A $ q$-difference analogue of $U({\mathfrak g})$ and the Yang–Baxter equations”, Lett. Math. Phys., 10 (1985), 63–69 | DOI | MR | Zbl

[48] Kulish P. P., Reshetikhin N. Yu., “Quantum linear problem for the sine-Gordon equation and higher representations”, J. Math. Sci., 23 (1983), 2435–2441 ; Reshetikhin N. Yu., Takhtadzhyan L. A., Faddeev L. D., “Quantization of Lie groups and Lie algebras”, Leningrad Math. J., 1 (1990), 193–225 | DOI | MR | Zbl

[49] Woronowicz S. L., “Compact matrix pseudogroups”, Comm. Math. Phys., 111 (1987), 613–665 | DOI | MR | Zbl

[50] Blohmann C., “Covariant realization of quantum spaces as star products by Drinfeld twists”, J. Math. Phys., 44 (2003), 4736–4755, arXiv: ; Blohmann C., “Realization of $q$-deformed spacetime as star product by a Drinfeld twist”, Proceedings of the 24th International Colloquium on Group Theoretical Methods in Physics (Paris, 2002), IOP Conference Series, 173, eds. J. P. Gazeau, R. Kerner, J. P. Antoine, S. Metens and J. Y. Thibon, 2003, 443–446, arXiv: ; Aizawa N., Chakrabarti R., “Noncommutative geometry of super-Jordanian $OSp_h(2/1)$ covariant quantum space”, J. Math. Phys., 45 (2004), 1623–1638, arXiv: math.QA/0209180math.QA/0402199math.QA/0311161 | DOI | MR | Zbl | DOI | MR | Zbl

[51] Lukierski J., “Deformed quantum relativistic phase spaces – an overview,”, Proceedings of III International Workshop “Classical and Quantum Integrable Systems” (Yerevan, 1998), eds. L. D. Mardoyan et al., JINR Dubna Publ. Dept., 1999, 141–152, arXiv: ; Kowalski-Glikman J., Nowak S., “Non-commutative space-time of doubly special relativity theories”, Internat. J. Modern Phys. D, 12 (2003), 299–315, arXiv: ; Granik A., Maguejo–Smolin transformation as a consequence of a specific definition of mass, velocity, and the upper limit on energy, arXiv: ; Mignemi S., “Transformations of coordinates and Hamiltonian formalism in deformed special relativity”, Phys. Rev. D, 68 (2003), 065029, 6 pp., arXiv: ; Ghosh S., “Lagrangian for doubly special relativity particle and the role of noncommutativity”, Phys. Rev. D, 74 (2006), 084019, 5 pp., arXiv: ; Ghosh S., Pal P., “Deformed special relativity and deformed symmetries in a canonical framework”, Phys. Rev. D, 75 (2007), 105021, 11 pp., arXiv: ; Antonio Garcia J., “Doubly special relativity and canonical transformations: Comment on “Lagrangian for doubly special relativity particle and the role of noncommutativity””, Phys. Rev. D, 76 (2007), 048501, 2 pp., arXiv: hep-th/9812063hep-th/0204245hep-th/0207113gr-qc/0304029hep-th/0608206hep-th/07021590705.0143 | DOI | MR | Zbl | DOI | MR | DOI | MR | DOI | MR | DOI | MR

[52] Frydryszak A. M., Tkachuk V. M., “Aspects of pre-quantum description of deformed theories”, Czechoslovak J. Phys., 53 (2003), 1035–1040 | DOI | MR

[53] Kowalski-Glikman J., “De Sitter space as an arena for doubly special relativity”, Phys. Lett. B, 547 (2002), 291–296, arXiv: hep-th/0207279 | DOI | MR | Zbl

[54] Kontsevich M., “Deformation quantization of Poisson manifolds”, Lett. Math. Phys., 66 (2003), 157–216, arXiv: q-alg/9709040 | DOI | MR | Zbl

[55] Oriti D., “Emergent non-commutative matter fields from group field theory models of quantum spacetime”, J. Phys. Conf. Ser., 174 (2009), 012047, 14 pp., arXiv: 0903.3970 | DOI

[56] Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | MR

[57] Kassel C., Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995 | MR | Zbl

[58] Majid S., “Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras”, Generalized Symmetries in Physics (Clausthal, 1993), World Sci. Publ., River Edge, NJ, 1994, 13–41, arXiv: hep-th/9311184 | MR

[59] Blattner R. J., Cohen M., Montgomery S., “Crossed products and inner actions of Hopf algebras”, Trans. Amer. Math. Soc., 298 (1986), 671–711 ; Blattner R. J., Montgomery S., “Crossed products and Galois extensions of Hopf algebras”, Pacific J. Math., 137 (1989), 37–54 ; Doi Y., Takeuchi M., “Cleft comodule algebras for a bialgebra”, Comm. Algebra, 14 (1986), 801–817 ; Doi Y., “Equivalent crossed products for a Hopf algebra”, Comm. Algebra, 17 (1989), 3053–3085 ; Cohen M., Fischman D., Montgomery S., “Hopf Galois extensions, smash products and Morita equivalence”, J. Algebra, 133 (1990), 351–372 ; Borowiec A., Marcinek W., “On crossed product of algebras”, J. Math. Phys., 41 (2000), 6959–6975, arXiv: math-ph/0007031 | DOI | MR | Zbl | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[60] Lu J.-H., “On the Drinfeld double and the Heisenberg double of a Hopf algebra”, Duke Math. J., 74 (1994), 763–776 ; Kashaev R. M., “The Heisenberg double and the pentagon relation”, St. Petersburg Math. J., 8 (1997), 585–592, arXiv: ; Skoda Z., Heisenberg double versus deformed derivatives, arXiv: q-alg/95030050909.3769 | DOI | MR | Zbl | MR

[61] Lukierski J., Minnaert P., Nowicki A., “$D=4$ quantum Poincaré–Heisenberg algebra”, Symmetries in Science, VI (Bregenz, 1992), eds. B. Gruber, Plenum, New York, 1993, 469–475 | MR | Zbl

[62] Stueckelberg E. C. G., “Remarque à propos de la création de paires de particules en théorie de relativité”, Helvetica Phys. Acta, 14 (1941), 588–594 ; Stueckelberg E. C. G., “La mécanique du point matériel en théorie de relativité et en théorie des quanta”, Helvetica Phys. Acta, 15 (1942), 23–37 ; Cooke J. H., “Proper-time formulation of quantum mechanics”, Phys. Rev., 166 (1968), 1293–1298 ; Johnson J. E., “Position operators and proper time in relativistic quantum mechanics”, Phys. Rev., 181 (1969), 1755–1764 ; Johnson J. E., “Proper-time quantum mechanics. II”, Phys. Rev. D, 3 (1971), 1735–1747 ; Broyles A. A., “Space-time position operators”, Phys. Rev. D, 1 (1970), 979–988 ; Aghassi J. J., Roman P., Santilli R. M., “New dynamical group for the relativistic quantum mechanics of elementary particles”, Phys. Rev. D, 1 (1970), 2753–2765 ; Mensky M. B., “Relativistic quantum theory without quantized fields. I. Particles in the Minkowski space”, Comm. Math. Phys., 47 (1976), 97–108 | MR | MR | DOI | DOI | MR | DOI | MR | DOI | MR | DOI | Zbl | DOI | MR | Zbl

[63] Mendes R. V., “Deformations, stable theories and fundamental constants”, J. Phys. A: Math. Gen., 27 (1994), 8091–8104 | DOI | MR | Zbl

[64] Chryssomalakos C., Okon E., “Generalized quantum relativistic kinematics: a stability point of view”, Internat. J. Modern Phys. D, 13 (2003), 2003–2034, arXiv: ; Gresnigt N. G., Renaud P. F., Butler P. H., “The stabilized Poincaré–Heisenberg algebra: a Clifford algebra viewpoint”, Internat. J. Modern Phys. D, 16 (2007), 1519–1529, arXiv: ; Ahluwalia-Khalilova D. V., Gresnigt N. G., Nielsen A. B., Schritt D., Watson T. F., “Possible polarization and spin-dependent aspects of quantum gravity”, Internat. J. Modern Phys. D, 17 (2008), 495–504, arXiv: hep-th/0410212hep-th/06110340704.1669 | DOI | MR | DOI | MR | Zbl | DOI | MR

[65] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[66] Fiore G., Steinacker H., Wess J., “Unbraiding the braided tensor product”, J. Math. Phys., 44 (2003), 1297–1321, arXiv: ; Fiore G., Steinacker H., Wess J., “Decoupling braided tensor factors”, Phys. Atomic Nuclei, 64 (2001), 2116–2120, arXiv: math.QA/0007174math.QA/0012199 | DOI | MR | Zbl | DOI | MR

[67] Bonneau P., Gerstenhaber M., Giaquinto A., Sternheimer D., “Quantum groups and deformation quantization: explicit approaches and implicit aspects”, J. Math. Phys., 45 (2004), 3703–3741 | DOI | MR | Zbl

[68] Neshveyev S., Tuset L., Notes on the Kazhdan–Lusztig theorem on equivalence of the Drinfel'd category and categories of $U_q(g)$-modules, arXiv: ; Vaes S., Vaĭnerman L., “On low-dimensional locally compact quantum groups”, Locally Compact Quantum Groups and Groupoids (Strasbourg, 2002), IRMA Lect. Math. Theor. Phys., 2, ed. L. Vaĭnerman, de Gruyter, Berlin, 2003, 127–187, arXiv: ; De Commer K., On the construction of quantum homogeneous spaces from $*$-Galois objects, arXiv: 0711.4302math.QA/02072711001.2153 | MR | Zbl

[69] Reshetikhin N. Yu., “Multiparametric quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys., 20 (1990), 331–335 | DOI | MR | Zbl

[70] Gerstenhaber M., Giaquinto A., Schack S. D., “Quantum symmetry”, Quantum Groups (Leningrad, 1990), Lecture Notes in Math., 1510, ed. P. P. Kulish, Springer, Berlin, 1992, 9–46 ; Ogievetsky O. V., “Hopf structures on the Borel subalgebra of $\rm sl(2)$”, Suppl. Rendic. Cir. Math. Palermo Ser. II, 37 (1993), 185–199 ; Giaquinto A., Zhang J. J., “Bialgebra actions, twists, and universal deformation formulas”, J. Pure Appl. Algebra, 128 (1998), 133–151, arXiv: hep-th/9411140 | MR | MR | DOI | MR | Zbl

[71] Kulish P. P., Lyakhovsky V. D., Mudrov A. I., “Extended jordanian twists for Lie algebras”, J. Math. Phys., 40 (1999), 4569–4586, arXiv: ; Lyakhovsky V. D., del Olmo M. A., “Peripheric extended twists”, J. Phys. A: Math. Gen., 32 (1999), 4541–4552, arXiv: ; Lyakhovsky V. D., del Olmo M. A., “Chains of twists and induced deformations”, Czechoslovak J. Phys., 50 (2000), 129–134 math.QA/9806014math.QA/9811153 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[72] Tolstoy V. N., “Chains of extended Jordanian twists for Lie superalgebras”, Supersymmetries and Quantum Symmetries (SQS'03) (Dubna, 2003), eds. E. Ivanov and A. Pashnev, Publ. JINR, Dubna, 2004, 242–251, arXiv: ; Tolstoy V. N., “Multiparameter quantum deformations of Jordanian type for Lie superalgebras”, Differential geometry and physics, Nankai Tracts Math., 10, World Sci. Publ., Hackensack, NJ, 2006, 443–452, arXiv: math.QA/0402433math.QA/0701079 | MR | Zbl

[73] Tolstoy V. N., “Twisted quantum deformations of Lorentz and Poincaré algebras”, Lie Theory and Its Applications in Physics (Varna, 2007), eds. H.-D. Doebner and V. K. Dobrev, Heron Press, Sofia, 2008, 441–459, arXiv: 0712.3962

[74] Lukierski J., Ruegg H., Tolstoy V. N., Nowicki A., “Twisted classical Poincaré algebras”, J. Phys. A: Math. Gen., 27 (1994), 2389–2399, arXiv: ; Borowiec A., Lukierski J., Tolstoy V. N., “Once again about quantum deformations of $D=4$ Lorentz algebra: twistings of $q$-deformation”, Eur. Phys. J. C, 57 (2008), 601–611, arXiv: ; 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, arXiv: ; 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, arXiv: ; Borowiec A., Lukierski J., Tolstoy V. N., “On twist quantizations of $D=4$ Lorentz and Poincaré algebras”, Czechoslovak J. Phys., 55 (2005), 1351–1356, arXiv: ; Borowiec A., Lukierski J., Tolstoy V. N., “Basic twist quantization of ${\rm osp}(1|2)$ and $\kappa$-deformation of $D=1$ superconformal mechanics”, Modern Phys. Lett. A, 18 (2003), 1157–1169, arXiv: hep-th/93120680804.3305hep-th/0604146hep-th/0412131hep-th/0510154hep-th/0301033 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[75] Zakrzewski S., “Poisson structures on the Lorentz group”, Lett. Math. Phys., 32 (1994), 11–23 ; Zakrzewski S., “Poisson structures on the Poincaré group”, Comm. Math. Phys., 187 (1997), 285–311, arXiv: q-alg/9602001 | DOI | MR | Zbl | DOI | MR

[76] Lyakhovsky V. D., “Twist deformations of $\kappa$-Poincaré algebra”, Rep. Math. Phys., 61 (2008), 213–220 ; Daszkiewicz M., “Generalized twist deformations of Poincaré and Galilei Hopf algebras”, Rep. Math. Phys., 63 (2009), 263–277, arXiv: 0812.1613 | DOI | MR | Zbl | DOI | MR | Zbl

[77] Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., “Deformation theory and quantization. I. Deformations of symplectic structures”, Ann. Physics, 111 (1978), 61–110 | DOI | MR | Zbl

[78] Fiore G., “Deforming maps for Lie group covariant creation and annihilation operators”, J. Math. Phys., 39 (1998), 3437–3452, arXiv: ; Fiore G., “Drinfeld twist and $q$-deforming maps for Lie group covariant Heisenberg algebrae”, Rev. Math. Phys., 12 (2000), 327–359, arXiv: q-alg/9610005q-alg/9708017 | DOI | MR | Zbl | DOI | MR | Zbl

[79] Sheng Y., “Linear Poisson structures on $\mathbb R^4$”, J. Geom. Phys., 57 (2007), 2398–2410, arXiv: 0707.2870 | DOI | MR | Zbl

[80] Kathotia V., “Kontsevich's universal formula for deformation quantization and the Campbell–Baker–Hausdorff formula”, Internat. J. Math., 11 (2000), 523–551, arXiv: math.QA/9811174 | DOI | MR | Zbl

[81] Beggs E. J., Majid S., Nonassociative Riemannian geometry by twisting, arXiv: ; Young C. A. S., Zegers R., “On $\kappa$-deformation and triangular quasibialgebra structure”, Nuclear Phys. B, 809 (2009), 439–451, arXiv: ; Young C. A. S., Zegers R., “Triangular quasi-Hopf algebra structures on certain non-semisimple quantum groups”, Comm. Math. Phys., 298 (2010), 585–611, arXiv: ; Balachandran A. P., Ibort A., Marmo G., Martone M., “Quantum fields on noncommutative spacetimes: theory and phenomenology”, SIGMA, 6 (2010), 052, 22 pp., arXiv: 0912.15530807.27450812.32571003.4356 | DOI | MR | Zbl | DOI | DOI

[82] Coleman S., Mandula J., “All possible symmetries of the $S$ matrix”, Phys. Rev., 159 (1967), 1251–1256 | DOI | Zbl

[83] Stachura P., “Towards a topological (dual of) quantum $\kappa$-Poincaré group”, Rep. Math. Phys., 57 (2006), 233–256, arXiv: hep-th/0505093 | DOI | MR | Zbl

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

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

[86] D'Andrea F., “Spectral geometry of $\kappa$-Minkowski space”, J. Math. Phys., 47 (2006), 062105, 19 pp., arXiv: ; Iochum B., Masson T., Schücker T., Sitarz A., Compact $\kappa$-deformation and spectral triples, arXiv: hep-th/05030121004.4190 | DOI | MR

[87] Albert J. et al., “Probing quantum gravity using photons from a flare of the active galactic nucleus Markarian 501 observed by the MAGIC telescope”, Phys. Lett. B, 668 (2008), 253–257, arXiv: 0708.2889 | DOI

[88] Abdo A. et al., “Fermi observations of high-energy gamma-ray emission from GRB 080916C”, Science, 323 (2009), 1688–1693 | DOI