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

Andrzej Borowiec ; Anna Pachoł

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

We extend our previous study of Hopf-algebraic $\kappa$-deformations of all inhomogeneous orthogonal Lie algebras $\mathrm{iso}(g)$ 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.

Détail

Keywords: 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.

Bibliographie

[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, arXiv: gr-qc/0312124 | DOI | MR

[2] 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

[3] Amelino-Camelia G., Smolin L., Starodubtsev A., “Quantum symmetry, the cosmological constant and {P}lanck-scale phenomenology”, Classical Quantum Gravity, 21 (2004), 3095–3110, arXiv: hep-th/0306134 | DOI | MR | Zbl

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

[5] Aschieri P., Castellani L., “Noncommutative gravity solutions”, J. Geom. Phys., 60 (2010), 375–393, arXiv: 0906.2774 | DOI | MR | Zbl

[6] 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

[7] 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

[8] Aschieri P., Schenkel A., “Noncommutative connections on bimodules and {D}rinfeld twist deformation”, Adv. Theor. Math. Phys., 18 (2014), 513–612, arXiv: 1210.0241 | DOI | MR

[9] Ballesteros A., Herranz F. J., del Olmo M. A., Santander M., “A new “null-plane” quantum {P}oincaré algebra”, Phys. Lett. B, 351 (1995), 137–145, arXiv: q-alg/9502019 | DOI | MR | Zbl

[10] Ballesteros Á., Herranz F. J., Meusburger C., “Drinfel'd doubles for {$(2+1)$}-gravity”, Classical Quantum Gravity, 30 (2013), 155012, 20 pp., arXiv: 1303.3080 | DOI | MR | Zbl

[11] Ballesteros A., Herranz F. J., Meusburger C., “A {$(2+1)$} non-commutative {D}rinfel'd double spacetime with cosmological constant”, Phys. Lett. B, 732 (2014), 201–209, arXiv: 1402.2884 | DOI | MR

[12] Ballesteros Á., Herranz F. J., Meusburger C., Naranjo P., “Twisted {$(2+1)$} {$\kappa$}-{A}d{S} algebra, {D}rinfel'd doubles and non-commutative spacetimes”, SIGMA, 10 (2014), 052, 26 pp., arXiv: 1403.4773 | DOI | MR | Zbl

[13] Ballesteros A., Herranz F. J., Pereña C. M., “Null-plane quantum universal {$R$}-matrix”, Phys. Lett. B, 391 (1997), 71–77, arXiv: q-alg/9607009 | DOI | MR

[14] Ballesteros A., Herranz F. J., Bruno N. R., “Quantum (anti)de Sitter algebras and generalizations of the kappa-Minkowski space”, Proceedings of 11th International Conference on Symmetry Methods in Physics (June 21–24, 2004, Prague), Joint Institute for Nuclear Research, Dubna, 2004, 1–20, arXiv: hep-th/0409295 | MR

[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 | DOI | MR | Zbl

[16] Bonneau P., Flato M., Pinczon G., “A natural and rigid model of quantum groups”, Lett. Math. Phys., 25 (1992), 75–84 | DOI | MR | Zbl

[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 pp., arXiv: 0912.3299 | DOI

[18] Borowiec A., Lukierski J., Pachoł A., “Twisting and $\kappa$-Poincaré”, J. Phys. A: Math. Theor., 47 (2014), 405203, 12 pp., arXiv: 1312.7807 | DOI | MR | Zbl

[19] Borowiec A., Lukierski J., Tolstoy V. N., “Jordanian quantum deformations of {$D=4$} anti-de {S}itter and {P}oincaré superalgebras”, Eur. Phys. J. C, 44 (2005), 139–145, arXiv: hep-th/0412131 | DOI | MR | Zbl

[20] Borowiec A., Lukierski J., Tolstoy V. N., “On twist quantizations of {$D=4$} {L}orentz and {P}oincaré algebras”, Czechoslovak J. Phys., 55 (2005), 1351–1356, arXiv: hep-th/0510154 | DOI | MR

[21] Borowiec A., Lukierski J., Tolstoy V. N., “Jordanian twist quantization of {$D=4$} {L}orentz and {P}oincaré algebras and {$D=3$} contraction limit”, Eur. Phys. J. C, 48 (2006), 633–639, arXiv: hep-th/0604146 | DOI | MR | Zbl

[22] Borowiec A., Lukierski J., Tolstoy V. N., “Quantum deformations of {$D=4$} {L}orentz algebra revisited: twistings of {$q$}-deformation”, Eur. Phys. J. C, 57 (2008), 601–611, arXiv: 0804.3305 | DOI | MR | Zbl

[23] 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

[24] Borowiec A., Pachoł A., “{$\kappa$}-{M}inkowski spacetimes and {DSR} algebras: fresh look and old problems”, SIGMA, 6 (2010), 086, 31 pp., arXiv: 1005.4429 | DOI | MR | Zbl

[25] Borowiec A., Pachoł A., “The classical basis for the {$\kappa$}-{P}oincaré {H}opf algebra and doubly special relativity theories”, J. Phys. A: Math. Theor., 43 (2010), 045203, 10 pp., arXiv: 0903.5251 | DOI | MR | Zbl

[26] Borowiec A., Pachoł A., “Unified description for $\kappa$-deformations of orthogonal groups”, Eur. Phys. J. C, 74 (2014), 2812, 9 pp., arXiv: 1311.4499 | DOI

[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, arXiv: hep-th/0107039 | DOI | Zbl

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

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

[30] D'Andrea F., “Spectral geometry of {$\kappa$}-{M}inkowski space”, J. Math. Phys., 47 (2006), 062105, 19 pp., arXiv: hep-th/0503012 | DOI | MR | Zbl

[31] Daszkiewicz M., “Generalized twist deformations of {P}oincaré and {G}alilei {H}opf algebras”, Rep. Math. Phys., 63 (2009), 263–277, arXiv: 0812.1613 | DOI | MR | Zbl

[32] Daszkiewicz M., Imiłkowska K., Kowalski-Glikman J., Nowak S., “Scalar field theory on {$\kappa$}-{M}inkowski space-time and doubly special relativity”, Internat. J. Modern Phys. A, 20 (2005), 4925–4940, arXiv: hep-th/0410058 | DOI | MR | Zbl

[33] Daszkiewicz M., Lukierski J., Woronowicz M., “{$\kappa$}-deformed statistics and classical four-momentum addition law”, Modern Phys. Lett. A, 23 (2008), 653–665, arXiv: hep-th/0703200 | DOI | MR | Zbl

[34] Daszkiewicz M., Lukierski J., Woronowicz M., “Towards quantum noncommutative {$\kappa$}-deformed field theory”, Phys. Rev. D, 77 (2008), 105007, 10 pp., arXiv: 0708.1561 | DOI | MR

[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 pp., arXiv: 0807.1992 | DOI | MR | Zbl

[36] de Graaf W. A., “Classification of solvable {L}ie algebras”, Experiment. Math., 14 (2005), 15–25, arXiv: math.RA/0404071 | DOI | MR | Zbl

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

[38] Drinfel'd V. G., “Quantum groups”, Proceedings of the International Congress of Mathematicians (Berkeley, Calif., 1986), v. 1, 2, Amer. Math. Soc., Providence, RI, 1987, 798–820 | MR

[39] Drinfel'd V. G., “Quasi-{H}opf algebras”, Leningrad Math. J., 1 (1989), 1419–1457 | MR

[40] Durhuus B., Sitarz A., “Star product realizations of {$\kappa$}-{M}inkowski space”, J. Noncommut. Geom., 7 (2013), 605–645, arXiv: 1104.0206 | DOI | MR | Zbl

[41] Freidel L., Kowalski-Glikman J., Nowak S., “Field theory on {$\kappa$}-{M}inkowski space revisited: {N}oether charges and breaking of {L}orentz symmetry”, Internat. J. Modern Phys. A, 23 (2008), 2687–2718, arXiv: 0706.3658 | DOI | MR | Zbl

[42] Govindarajan T. R., Gupta K. S., Harikumar E., Meljanac S., Meljanac D., “Twisted statistics in {$\kappa$}-{M}inkowski spacetime”, Phys. Rev. D, 77 (2008), 105010, 6 pp., arXiv: 0802.1576 | DOI | MR

[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 pp., arXiv: 0903.2355 | DOI | MR

[44] Gupta K. S., Meljanac S., Samsarov A., “Quantum statistics and noncommutative black holes”, Phys. Rev. D, 85 (2012), 045029, 8 pp., arXiv: 1108.0341 | DOI

[45] Harikumar E., Jurić T., Meljanac S., “Geodesic equation in $\kappa$-Minkowski spacetime”, Phys. Rev. D, 86 (2012), 045002, 8 pp., arXiv: 1203.1564 | DOI

[46] Hossenfelder S., “Minimal length scale scenarios for quantum gravity”, Living Rev. Relativity, 16 (2013), 2, 90 pp., arXiv: 1203.6191 | DOI

[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 | DOI | MR | Zbl

[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 | DOI

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

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

[51] Kosiński P., Maślanka P., “The {$\kappa$}-{W}eyl group and its algebra”, From Field Theory to Quantum Groups, World Sci. Publ., River Edge, NJ, 1996, 41–51, arXiv: q-alg/9512018 | Zbl

[52] Kovačević D., Meljanac S., “Kappa-{M}inkowski spacetime, kappa-{P}oincaré {H}opf algebra and realizations”, J. Phys. A: Math. Theor., 45 (2012), 135208, 24 pp., arXiv: 1110.0944 | DOI | MR | Zbl

[53] Kowalski-Glikman J., “Observer-independent quantum of mass”, Phys. Lett. A, 286 (2001), 391–394, arXiv: hep-th/0102098 | DOI | MR | Zbl

[54] Kulish P. P., Lyakhovsky V. D., Mudrov A. I., “Extended {J}ordanian twists for {L}ie algebras”, J. Math. Phys., 40 (1999), 4569–4586, arXiv: math.QA/9806014 | DOI | MR | Zbl

[55] Lukierski J., Lyakhovsky V. D., “Two-parameter extensions of the {$\kappa$}-{P}oincaré quantum deformation”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemp. Math., 391, Amer. Math. Soc., Providence, RI, 2005, 281–288, arXiv: hep-th/0406155 | DOI | MR

[56] Lukierski J., Lyakhovsky V. D., Mozrzymas M., “{$\kappa$}-deformations of {$D=4$} {W}eyl and conformal symmetries”, Phys. Lett. B, 538 (2002), 375–384, arXiv: hep-th/0203182 | DOI | MR | Zbl

[57] Lukierski J., Nowicki A., Ruegg H., “New quantum {P}oincaré algebra and {$\kappa$}-deformed field theory”, Phys. Lett. B, 293 (1992), 344–352 | DOI | MR | Zbl

[58] Lukierski J., Ruegg H., Nowicki A., Tolstoy V. N., “{$q$}-deformation of {P}oincaré algebra”, Phys. Lett. B, 264 (1991), 331–338 | DOI | MR

[59] 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

[60] Lukierski J., Woronowicz M., “New {L}ie-algebraic and quadratic deformations of {M}inkowski space from twisted {P}oincaré symmetries”, Phys. Lett. B, 633 (2006), 116–124, arXiv: hep-th/0508083 | DOI | MR | Zbl

[61] Lyakhovsky V. D., “Twist deformations of {$\kappa$}-{P}oincaré algebra”, Rep. Math. Phys., 61 (2008), 213–220 | DOI | MR | Zbl

[62] 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

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

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

[65] Matschull H.-J., Welling M., “Quantum mechanics of a point particle in {$(2+1)$}-dimensional gravity”, Classical Quantum Gravity, 15 (1998), 2981–3030, arXiv: gr-qc/9708054 | DOI | MR | Zbl

[66] 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

[67] Meljanac S., Samsarov A., “Scalar field theory on {$\kappa$}-{M}inkowski space-time and translation and {L}orentz invariance”, Internat. J. Modern Phys. A, 26 (2011), 1439–1468, arXiv: 1007.3943 | DOI | MR | Zbl

[68] Meljanac S., Samsarov A., Stojić M., Gupta K. S., “{$\kappa$}-{M}inkowski spacetime and the star product realizations”, Eur. Phys. J. C, 53 (2008), 295–309, arXiv: 0705.2471 | DOI | MR | Zbl

[69] Meljanac S., Samsarov A., Trampetić J., Wohlgenannt M., “Scalar field propagation in the $\phi^4$ kappa-Minkowski model”, J. High Energy Phys., 2011:12 (2011), 010, 23 pp., arXiv: 1111.5553 | DOI | MR

[70] Mercati F., Sitarz A., “$\kappa$-Minkowski differential calculi and star product”, PoS(CNCFG2010), PoS Proc. Sci., 2010, 030, 11 pp., arXiv: 1105.1599

[71] Meusburger C., Schroers B. J., “Generalised {C}hern–{S}imons actions for 3d gravity and {$\kappa$}-{P}oincaré symmetry”, Nuclear Phys. B, 806 (2009), 462–488, arXiv: 0805.3318 | DOI | MR | Zbl

[72] Mubarakzjanov G. M., “On solvable {L}ie algebras”, Izv. Vys. Ucheb. Zaved. Matematika, 1963, no. 1(32), 114–123 | MR

[73] Mudrov A. I., “Twisting cocycle for null-plane quantized Poincaré algebra”, J. Phys. A: Math. Gen., 31 (1998), 6219–6224, arXiv: q-alg/9711001 | DOI | MR | Zbl

[74] Ohl T., Schenkel A., “Cosmological and black hole spacetimes in twisted noncommutative gravity”, J. High Energy Phys., 2009:10 (2009), 052, 12 pp., arXiv: 0906.2730 | DOI | MR

[75] 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

[76] Patera J., Sharp R. T., Winternitz P., Zassenhaus H., “Invariants of real low dimension {L}ie algebras”, J. Math. Phys., 17 (1976), 986–994 | DOI | MR | Zbl

[77] Podleś P., Woronowicz S. L., “On the classification of quantum {P}oincaré groups”, Comm. Math. Phys., 178 (1996), 61–82, arXiv: hep-th/9412059 | DOI | MR | Zbl

[78] Podleś P., Woronowicz S. L., “On the structure of inhomogeneous quantum groups”, Comm. Math. Phys., 185 (1997), 325–358, arXiv: hep-th/9412058 | DOI | MR | Zbl

[79] Popovych R. O., Boyko V. M., Nesterenko M. O., Lutfullin M. W., “Realizations of real low-dimensional {L}ie algebras”, J. Phys. A: Math. Gen., 36 (2003), 7337–7360, arXiv: math-ph/0301029 | DOI | MR | Zbl

[80] Schenkel A., Uhlemann C. F., “Field theory on curved noncommutative spacetimes”, SIGMA, 6 (2010), 061, 19 pp., arXiv: 1003.3190 | DOI | MR | Zbl

[81] Sitarz A., “Twists and spectral triples for isospectral deformations”, Lett. Math. Phys., 58 (2001), 69–79, arXiv: math.QA/0102074 | DOI | MR | Zbl

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

[83] Tolstoy V. N., in honour of Jerzy Lukierski (September 27–29, 2006, Wroclaw, Poland), arXiv: 0704.0081

[84] Tolstoy V. N., “Twisted quantum deformations of Lorentz and Poincaré algebras”, {I}nvited 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 {P}oincaré algebra”, Nuclear Phys. B, 797 (2008), 537–549, arXiv: 0711.2206 | DOI | MR | Zbl

[86] Young C. A. S., Zegers R., “Deformation quasi-{H}opf algebras of non-semisimple type from cochain twists”, Comm. Math. Phys., 298 (2010), 585–611, arXiv: 0812.3257 | DOI | MR | Zbl

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

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