zbMATH — the first resource for mathematics

Generalized Heisenberg algebra applied to realizations of the orthogonal, Lorentz, and Poincaré algebras and their dual extensions. (English) Zbl 1443.81044
Summary: We introduce the generalized Heisenberg algebra \(\mathcal{H}_n\) and construct realizations of the orthogonal and Lorentz algebras by a formal power series in a semicompletion of \(\mathcal{H}_n\). The obtained realizations are given in terms of the generating function for the Bernoulli numbers. We also introduce an extension of the orthogonal and Lorentz algebras by quantum angles and study realizations of the extended algebras in \(\mathcal{H}_n\). Furthermore, we show that by extending the generalized Heisenberg algebra \(\mathcal{H}_n\), one can also obtain realizations of the Poincaré algebra and its extension by quantum angles.
©2020 American Institute of Physics
81R60 Noncommutative geometry in quantum theory
33C65 Appell, Horn and Lauricella functions
14D15 Formal methods and deformations in algebraic geometry
11R60 Cyclotomic function fields (class groups, Bernoulli objects, etc.)
22E43 Structure and representation of the Lorentz group
Full Text: DOI
[1] Doplicher, S.; Fredenhagen, K.; Roberts, J. E., Spacetime quantization induced by classical gravity, Phys. Lett. B, 331, 39-44 (1994)
[2] Doplicher, S.; Fredenhagen, K.; Roberts, J. E., The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172, 187-220 (1995) · Zbl 0847.53051
[3] Lukierski, J.; Ruegg, H.; Nowicki, A.; Tolstoy, V. N., q-deformation of Poincaré algebra, Phys. Lett. B, 264, 331-338 (1991)
[4] Majid, S.; Ruegg, H., Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B, 334, 348-354 (1994) · Zbl 1112.81328
[5] Kowalski-glikman, J.; Nowak, S., Noncommutative space-time of doubly special relativity theories, Int. J. Mod. Phys. D, 12, 299-315 (2003) · Zbl 1079.83535
[6] Kowalski-Glikman, J., Introduction to doubly special relativity, Lect. Notes Phys., 669, 131-159 (2005)
[7] Amelino-Camelia, G.; Lukierski, J.; Nowicki, A., Kappa-deformed covariant phase space and quantum gravity uncertainty relations, Phys. . At. Nucl., 61, 1811-1815 (1998) · Zbl 0988.81052
[8] Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., Towards quantum noncommutative κ-deformed field theory, Phys. Rev. D, 77, 105007 (2008)
[9] Govindarajan, T. R.; Gupta, K. S.; Harikumar, E.; Meljanac, S.; Meljanac, D., Deformed oscillator algebras and QFT in the κ-Minkowski spacetime, Phys. Rev. D, 80, 025014 (2009)
[10] Govindarajan, T. R.; Gupta, K. S.; Harikumar, E.; Meljanac, S.; Meljanac, D., Twisted statistics in κ-Minkowski spacetime, Phys. Rev. D, 77, 105010 (2008)
[11] Durov, N.; Meljanac, S.; Samsarov, A.; Škoda, Z., A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra, 309, 318-359 (2007) · Zbl 1173.17014
[12] Meljanac, S.; Stojić, M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C, 47, 531-539 (2006) · Zbl 1191.81138
[13] Meljanac, S.; Krešić-Jurić, S.; Stojić, M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C, 51, 229-240 (2007) · Zbl 1189.81114
[14] Meljanac, S.; Krešić-Jurić, S., Generalized kappa-deformed spaces, star products and their realizations, J. Phys. A: Math. Theor., 41, 235203 (2008) · Zbl 1140.81419
[15] Meljanac, S.; Samsarov, A.; Stojić, M.; Gupta, K. S., κ-Minkowski space-time and the star product realizations, Eur. Phys. J. C, 53, 295-309 (2008) · Zbl 1189.81115
[16] Meljanac, S.; Škoda, Z.; Svrtan, D., Exponential formulas and Lie algebra type star products, SIGMA, 8, 013 (2012) · Zbl 1248.81092
[17] Meljanac, S.; Meljanac, D.; Samsarov, A.; Stojić, M., κ-deformed Snyder spacetime, Mod. Phys. Lett. A, 25, 579-590 (2010) · Zbl 1188.83068
[18] Meljanac, S.; Meljanac, D.; Mignemi, S.; Štrajn, R., Snyder-type spaces, twisted Poincaré algebra and addition of momenta, Int. J. Mod. Phys. A, 32, 1750172 (2017) · Zbl 1376.81044
[19] Meljanac, S.; Meljanac, D.; Mercati, F.; Pikutić, D., Noncommutative spaces and Poincaré symmetry, Phys. Lett. B, 766, 181-185 (2017) · Zbl 1397.81106
[20] Meljanac, S.; Meljanac, D.; Pachoł, A.; Pikutić, D., Remarks on simple interpolation between Jordanian twists, J. Phys. A: Math. Theor., 50, 265201 (2017) · Zbl 1370.81094
[21] Meljanac, D.; Meljanac, S.; Pikutić, D.; Gupta, K. S., Twisted statistics and the structure of Lie-deformed Minkowski spaces, Phys. Rev. D, 96, 105008 (2017)
[22] Meljanac, D.; Meljanac, S.; Mignemi, S.; Štrajn, R., κ-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra and dispersion relations, Phys. Rev. D, 99, 126012 (2019)
[23] Harikumar, E.; Jurić, T.; Meljanac, S., Geodesic equation in κ-Minkowski spacetime, Phys. Rev. D, 86, 045002 (2012)
[24] Meljanac, S.; Krešić-Jurić, S.; Martinić, T., Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces, J. Math. Phys., 58, 071701 (2017) · Zbl 1370.83064
[25] Meljanac, S.; Krešić-Jurić, S.; Martinić, T., The Weyl realizations of Lie algebras, and left-right duality, J. Math. Phys., 57, 5, 051704 (2016) · Zbl 1342.58005
[26] Meljanac, S.; Krešić-Jurić, S.; Pikutić, D., Generalization of Weyl realization to a class of Lie superalgebras, J. Math. Phys., 59, 021701 (2018) · Zbl 1390.17010
[27] Zakrzewski, S., Quantum Poincaré group related to κ-Poincaré algebra, J. Phys. A: Math. Gen., 27, 2075 (1994) · Zbl 0834.17024
[28] Lukierski, J.; Meljanac, S.; Woronowicz, M., Quantum twist-deformed D = 4 phase spaces with spin sector and Hopf algebroid structures, Phys. Lett. B, 789, 82-87 (2019) · Zbl 1406.81050
[29] Lukierski, J.; Meljanac, D.; Meljanac, S.; Pikutić, D.; Woronowicz, M., Lie-deformed quantum Mikonwski spaces from twists: Hopf algebraic versus Hopf-algebroid approach, Phys. Lett. B, 777, 1-7 (2018) · Zbl 1411.81119
[30] Lukierski, J.; Škoda, Z.; Woronowicz, M., Deformed covariant quantum phase spaces as Hopf algebroids, Phys. Lett. B, 750, 401-406 (2015) · Zbl 1364.81170
[31] Lukierski, J.; Woronowicz, M., New Lie-algebraic and quadratic deformations of Minkowski space from twisted Poincaré symmetries, Phys. Lett. B, 633, 116-124 (2006) · Zbl 1247.81216
[32] Kosinski, P.; Maslanka, P., The duality between κ-Poincaré algebra and κ-Poincaré group · Zbl 1058.81558
[33] Lukierski, J.; Nowicki, A.; Dobrev, V. K.; Doebner, H. D., Heisenberg double description of κ-Poincare algebra and κ-deformed phase space, 186 (1997), Heron Press: Heron Press, Sofia
[34] Fadeev, L. D.; Reshetikhin, N. Yu.; Takhtajan, L.; Kashiwara, M.; Kawai, T., Quantization of Lie groups and Lie algebras, Algebraic Analysis, 129-139 (1990), Academic Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.