zbMATH — the first resource for mathematics

The Weyl realizations of Lie algebras, and left-right duality. (English) Zbl 1342.58005
Summary: We investigate dual realizations of non-commutative spaces of Lie algebra type in terms of formal power series in the Weyl algebra. To each realization of a Lie algebra we associate a star-product on the symmetric algebra \(S(\mathfrak{g})\) and an ordering on the enveloping algebra \(U(\mathfrak{g})\). Dual realizations of are defined in terms of left-right duality of the star-products on \(S(\mathfrak{g})\). It is shown that the dual realizations are related to an extension problem for \(\mathfrak{g}\) by shift operators whose action on \(U(\mathfrak{g})\) describes left and right shift of the generators of \(U(\mathfrak{g})\) in a given monomial. Using properties of the extended algebra, in the Weyl symmetric ordering we derive closed form expressions for the dual realizations of \(\mathfrak{g}\) in terms of two generating functions for the Bernoulli numbers. The theory is illustrated by considering the \(\kappa\)-deformed space.
©2016 American Institute of Physics

58B34 Noncommutative geometry (à la Connes)
11L15 Weyl sums
53D55 Deformation quantization, star products
11B68 Bernoulli and Euler numbers and polynomials
Full Text: DOI arXiv
[1] Olver, P. J., Applications of Lie Groups to Differential Equations, (1998), Springer-Verlag: Springer-Verlag, New York
[2] Ovsiannikov, L. V., Group Analysis of Differential Equations, (1982), Academic Press: Academic Press, New York · Zbl 0485.58002
[3] Basarab-Horwath, P.; Lahno, V.; Zhdanov, R., The structure of Lie algebras and the classification problem for partial differential equations, Acta Appl. Math., 69, 1, 43-94, (2001) · Zbl 1054.35002
[4] Bourlioux, A.; Cyr-Gagnon, C.; Winternitz, P., Difference schemes with point symmetries and their numerical tests, J. Math. Phys. A: Math. Gen., 39, 6877-6896, (2006) · Zbl 1095.65070
[5] Doplicher, S.; Fredenhagen, K.; Roberts, J., Spacetime quantization induced by classical gravity, Phys. Lett. B, 331, 39-44, (1994)
[6] 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
[7] Seiberg, N.; Witten, E., String theory and noncommutative geometry, JHEP, 09, 032, (1999) · Zbl 0957.81085
[8] Moyal, J. E., Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc., 45, 99-124, (1949) · Zbl 0031.33601
[9] Szabo, R. J., Symmetry, gravity and noncommutativity, Classical Quantum Gravity, 23, R199-R242, (2006) · Zbl 1117.83001
[10] Lukierski, J.; Nowicki, A.; Ruegg, H., q-deformation of Poincaré algebra, Phys. Lett. B, 264, 331-338, (1991)
[11] Majid, S.; Ruegg, H., Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B, 334, 348-354, (1994) · Zbl 1112.81328
[12] 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
[13] Li, M.; Wu, Y. S., Physics in Noncommutative World: Field Theories, (2002), Rinton Press: Rinton Press, Princenton
[14] Aschieri, P.; Dimitrijević, M.; Kulish, P.; Lizzi, F.; Wess, J., Noncommutative Spacetimes: Symmetry in Noncommutative Geometry and Field Theory, 774, (2009), Springer: Springer, Berlin · Zbl 1177.81004
[15] Jurić, T.; Kovačević, D.; Meljanac, S., κ-deformed phase space, Hopf algebroid and twisting, SIGMA, 10, 106, (2014) · Zbl 1310.81098
[16] Meljanac, S.; Škoda, Z.; Stojić, M., Lie algebra type noncommutative phase spaces are Hopf algebroids, (2014)
[17] Durov, N.; Meljanac, S.; Samsarov, A.; Škoda, Z., A universal formula for representing Lie aglebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra, 309, 318-359, (2007) · Zbl 1173.17014
[18] Kovačević, D.; Meljanac, S.; Samsarov, A.; Škoda, Z., Hermitian realizations of κ-Minkowski space-time, Int. J. Mod. Phys. A, 30, 1550019, (2015) · Zbl 1310.81099
[19] Meljanac, S.; Stojić, M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C, 47, 531-539, (2006) · Zbl 1191.81138
[20] Meljanac, S.; Krešić-Jurić, S.; Stojić, M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C, 51, 229-240, (2007) · Zbl 1189.81114
[21] Meljanac, S.; Škoda, Z.; Svrtan, D., Exponential formulas and Lie algebra type star products, SIGMA, 8, 013, (2012) · Zbl 1248.81092
[22] Meljanac, S.; Samsarov, A.; Stojić, M.; Gupta, K. S., Kappa-Minkowski space-time and the star product realizations, Eur. Phys. J. C, 53, 295-309, (2008) · Zbl 1189.81115
[23] Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216, (2003) · Zbl 1058.53065
[24] Gutt, S., An explicit ⋆-product on the cotangent bundle of a Lie group, Lett. Math. Phys., 7, 249-258, (1983) · Zbl 0522.58019
[25] 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
[26] Kowalski-Glikman, J., Introduction to doubly special relativity, Lect. Notes Phys., 669, 131-159, (2005)
[27] 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
[28] Daszkiewicz, M.; Lukierski, J.; Woronowicz, M., Towards quantum noncommutative κ-deformed field theory, Phys. Rev. D, 77, 105007, (2008)
[29] 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)
[30] Meljanac, S.; Krešić-Jurić, S., Noncommutative differential forms on the κ-deformed space, J. Phys. A: Math. Theor., 42, 365204, (2009) · Zbl 1202.58004
[31] Meljanac, S.; Krešić-Jurić, S., Differential structure on κ-Minkowski spacetime, and κ-Poincaré algebra, Int. J. Mod. Phys. A, 26, 20, 3385-3402, (2011) · Zbl 1247.81642
[32] Meljanac, S.; Krešić-Jurić, S.; Štrajn, R., Differential algebra on κ-Minkowski space and action of the Lorentz algebra, Int. J. Mod. Phys. A, 27, 10, 1250057, (2012) · Zbl 1247.83009
[33] Jurić, T.; Meljanac, S.; Štrajn, R., Universal κ-Poincaré covariant differential calculus over κ-Minkowski space, Int. J. Mod. Phys. A, 29, 1450121, (2014) · Zbl 1297.81111
[34] Jurić, T.; Meljanac, S.; Pikutić, D.; Štrajn, R., Towards the classification of differential calculi on κ-Minkowski space and related field theories, JHEP, 1507, 055, (2015) · Zbl 1388.83537
[35] Kovačević, D.; Meljanac, S., Kappa-Minkowski spacetime, kappa-Poincaré algebra and realizations, J. Phys. A: Math. Theor., 45, 135208, (2012) · Zbl 1241.83010
[36] Kathotia, V., Kontsevich’s universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, Int. J. Math., 11, 4, 523-551, (2000) · Zbl 1110.53308
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.