×

zbMATH — the first resource for mathematics

Realization of bi-covariant differential calculus on the Lie algebra type noncommutative spaces. (English) Zbl 1370.83064
Summary: This paper investigates bi-covariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra \(\mathfrak{g}_0\), we construct a Lie superalgebra \(\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1\) containing noncommutative coordinates and one-forms. We show that \(\mathfrak{g}\) can be extended by a set of generators \(T_{AB}\) whose action on the enveloping algebra \(U(\mathfrak{g})\) gives the commutation relations between monomials in \(U(\mathfrak{g}_0)\) and one-forms. Realizations of noncommutative coordinates, one-forms, and the generators \(T_{AB}\) as formal power series in a semicompleted Weyl superalgebra are found. In the special case \(\dim(\mathfrak{g}_0) = \dim(\mathfrak{g}_1)\), we also find a realization of the exterior derivative on \(U(\mathfrak{g}_0)\). The realizations of these geometric objects yield a bi-covariant differential calculus on \(U(\mathfrak{g}_0)\) as a deformation of the standard calculus on the Euclidean space.
©2017 American Institute of Physics

MSC:
83C65 Methods of noncommutative geometry in general relativity
17A70 Superalgebras
17B35 Universal enveloping (super)algebras
83C45 Quantization of the gravitational field
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Doplicher, S.; Fredenhagen, K.; Roberts, J., Spacetime quantization induced by classical gravity, Phys. Lett. B, 331, 39-44, (1994)
[2] Doplicher, S.; Fredenhagen, K.; Roberts, J., The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172, 187-220, (1995) · Zbl 0847.53051
[3] Woronowicz, S. L., Differential calculus on compact matrix groups (quantum groups), Commun. Math. Phys., 122, 125-170, (1989) · Zbl 0751.58042
[4] Schupp, P.; Watts, P.; Zumino, B., Differential geometry on linear quantum groups, Lett. Math. Phys., 25, 139-147, (1992) · Zbl 0765.17020
[5] Radtko, O. V.; Vladimirov, A. A., On the algebraic structure of differential calculus on quantum groups, J. Math. Phys., 38, 10, 5434, (1997) · Zbl 0902.16033
[6] Moyal, J. E., Quantum mechanics as a statistical theory, Math. Proc. Cambridge Philos. Soc., 45, 99-124, (1949) · Zbl 0031.33601
[7] Groenewold, H. J., On the principles of elementary quantum mechanics, Physica, 12, 7, 405-460, (1946) · Zbl 0060.45002
[8] Lukierski, J.; Nowicki, A.; Ruegg, H., q-deformation of Poincaré algebra, Phys. Lett. B, 264, 331-338, (1992)
[9] Zakrzewski, S., Quantum Poincaré group related to the \(\kappa\)-Poincaré algebra, J. Phys. A: Math. Gen., 27, 2075-2082, (1994) · Zbl 0834.17024
[10] Majid, S.; Ruegg, H., Bicrossproduct structure of κ-Poincarće group and non-commutative geometry, Phys. Lett. B, 334, 348-354, (1994) · Zbl 1112.81328
[11] Snyder, H. S., Quantized spacetime, Phys. Rev., 71, 1, 38-41, (1947) · Zbl 0035.13101
[12] Li, M.; Wu, Y. S., Physics in Noncommutative World: Field Theories, (2002), Rinton Press: Rinton Press, Princeton
[13] Aschieri, P.; Dimitrijević, M.; Kulish, P.; Lizzi, F.; Wess, J., Noncommutative Spacetimes: Symmetry in Noncommutative Geometry and Field Theory, (2009), Springer: Springer, Berlin · Zbl 1177.81004
[14] Landi, G., An Introduction to Noncommutative Spaces and Their Geometry, Lecture Notes in Physics Monographs, (2002), Springer: Springer, Berlin
[15] Sitarz, A., Noncommutative differential calculus on the \(\kappa\)-Minkowski space, Phys. Lett. B, 349, 42-48, (1995)
[16] Gonera, C.; Kosinski, P.; Maslanka, P., Differential calculi on quantum Minkowski space, J. Math. Phys., 37, 11, 5820, (1996) · Zbl 0865.17010
[17] Mercati, F., Quantum \(\kappa\)-deformed differential geometry and field theory, Int. J. Mod. Phys. D, 25, 5, 1650053, (2016) · Zbl 1338.81218
[18] Jurić, T.; Meljanac, S.; Pikutić, D.; Štrajn, R., Toward the classification of differential calculi on \(\kappa\)-Minkowski space and related field theories, J. High Energy Phys., 07, 055, (2015) · Zbl 1388.83537
[19] 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
[20] Meljanac, S.; Krešić-Jurić, S., Noncommutative differential forms on the kappa-deformed space, J. Phys. A: Math. Theor., 42, 36, 365204, (2009) · Zbl 1202.58004
[21] Meljanac, S.; Krešić-Jurić, S., Differential structure on \(\kappa\)-Minkowski space, and \(\kappa\)-Poincaré algebra, Int. J. Mod. Phys. A, 26, 20, 3385-3402, (2011) · Zbl 1247.81642
[22] Meljanac, S.; Krešić-Jurić, S.; Štrajn, R., Differential algebras on \(\kappa\)-Minkowski space and action of the Lorentz algebra, Int. J. Mod. Phys. A, 27, 10, 1250057, (2012) · Zbl 1247.83009
[23] Jurić, T.; Meljanac, S.; Štrajn, R., Diffeerential forms on \(\kappa\)-Minkowski spacetime from extended twist, Eur. Phys. J. C, 73, 2472, (2013) · Zbl 1311.81155
[24] Jurić, T.; Meljanac, S.; Štrajn, R., Universal \(\kappa\)-Poincaré differential calculus over \(\kappa\)-Minkowski space, Int. J. Mod. Phys. A, 29, 1450121, (2014) · Zbl 1297.81111
[25] 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
[26] Klimyk, A.; Schmüdgen, K., Quantum Groups and Their Representations, (1997), Springer: Springer, Berlin · Zbl 0891.17010
[27] Kowalski-Glikman, J.; Nowak, S., Non-commutative space-time of doubly special relativity theories, Int. J. Mod. Phys. D, 12, 299-315, (2003) · Zbl 1079.83535
[28] Amelino-Camelia, G.; Lukierski, J.; Nowicki, A., Kappa-deformed covariant phase space and quantum gravity uncertainty relations, Phys. Atom Nucl., 61, 1811-1815, (1998) · Zbl 0988.81052
[29] Govindarajan, T. R.; Gupta, K. S.; Harikumar, E.; Meljanac, S.; Meljanac, D., Deformed oscillator algebras and QFT in the \(\kappa\)-Minkowski spacetime, Phys. Rev. D, 80, 025014, (2009)
[30] Meljanac, S.; Stojić, M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C, 47, 531-539, (2006) · Zbl 1191.81138
[31] Meljanac, S.; Krešić-Jurić, S.; Stojić, M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C, 51, 229-240, (2007) · Zbl 1189.81114
[32] Meljanac, S.; Meljanac, D.; Mercati, F.; Pikutić, D., Noncommutative spaces and Poincaré symmetry, Phys. Lett. B, 766, 181-185, (2017) · Zbl 1397.81106
[33] Govindarajan, T. R.; Gupta, K. S.; Harikumar, E.; Meljanac, S.; Meljanac, D., Twisted statistics in \(\kappa\)-Minkowski spacetime, Phys. Rev. D, 77, 105010, (2008)
[34] Jurić, T.; Meljanac, S.; Štrajn, R., Twists, realizations and Hopf algebroid structure of \(\kappa\)-deformed phase space, Int. J. Mod. Phys. A, 29, 5, 1450022, (2014) · Zbl 1284.81175
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.