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On the quantum Lorentz group. (English) Zbl 1145.81382
Summary: The quantum analog of Pauli matrices are introduced and investigated. From these matrices and an appropriate trace over spinorial indices we construct a quantum Minkowski metric. In this framework we show explicitly the correspondence between the \(\text{SL}(2,\mathbb C)\) and Lorentz quantum groups. Five Image matrices of the quantum Lorentz group are constructed in terms of the R matrix of \(\text{SL}(2,\mathbb C)\) group. These Image matrices satisfy Yang-Baxter equations and two of which have adequate properties tied to the quantum Minkowski space structure as the reality conditions of the coordinates and the symmetrization of the metric. It is also shown that the Minkowski metric leads to invariant and central lengths of four-vectors.
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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[1] Connes, A., Publ. IHES, 62, 257, (1986)
[2] Coquereaux, R., Noncommutative geometry and theoretical physics, J. geom. phys., 6, 425-490, (1989) · Zbl 0695.16020
[3] Dubois-Violette, M.; Kerner, R.; Madore, J., Noncommutative differential geometry of matrix algebras, J. math. phys., 31, 316-322, (1990) · Zbl 0704.53081
[4] Woronowicz, S.L., Differential calculus on compact matrix pseudogroups, Commun. math. phys., 122, 125-170, (1989) · Zbl 0751.58042
[5] Podels, P.; Woronowicz, S.L., Quantum deformation of Lorentz group, Commun. math. phys., 130, 381-431, (1990) · Zbl 0703.22018
[6] Podles, P.; Woronowicz, S.L., On the classification of quantum Poincaré group, Commun. math. phys., 178, 61-82, (1996) · Zbl 0869.17010
[7] Carow-Watamura, U.; Schlieker, M.; Scholl, M.; Watamura, S., Tensor representation of the quantum group SLq(2,C) and quantum Minkowski space, Z. phys. C, 48, 159, (1990)
[8] Schmidke, W.B.; Wess, J.; Zumino, B., A q-deformed Lorentz algebra, Z. phys. C, 57, 495-518, (1991) · Zbl 0793.17005
[9] Ogiesvestsky, O.; Schmidke, W.B.; Wess, J.; Zumino, B., q-deformed Poincaré algebra, Commun. math. phys., 150, 495-518, (1992) · Zbl 0849.17011
[10] Pillin, M.; Weikl, L., On representation of the q-deformed Lorentz and Poincaré algebras, J. phys. A, 27, 5525-5540, (1994) · Zbl 0834.17020
[11] Dubois-Violette, M.; Launer, G., The quantum group of a non-degenerate bilinear form, Phys. lett. B, 245, 175-178, (1990) · Zbl 1119.16307
[12] Lagraa, M., Quantum gauge theories, Int. J. mod. phys. A, 11, 699-713, (1996) · Zbl 0985.81550
[13] Hammou, A.B.; Lagraa, M., The B.R.S.T. operator of quantum symmetries, the quantum analog of the Donaldson invariant, J. math. phys., 38, 4462-4472, (1997) · Zbl 0910.53049
[14] M. Lagraa, On the measurability of observables in noncommutative special relativity, Math.ph/9904014.
[15] A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.
[16] Coquereaux, R.; Esposito-Farese, G.; Vaillant, G., Higgs fields as yang – mills fields and discrete symmetries, Nucl. phys. B, 353, 689-706, (1991)
[17] Dubois-Violette, M.; Kerner, R.; Madore, J., Noncommutative differential geometry and new models of gauge theory, J. math. phys., 31, 323-330, (1990) · Zbl 0704.53082
[18] Carow-Watamura, U.; Schlieker, M.; Scholl, M.; Watamura, S., A quantum Lorentz group, Int. J. mod. phys. A, 6, 3081-3108, (1991) · Zbl 0736.17018
[19] Benaoum, H.B.; Lagraa, M., Uq(2) yang – mills theory, Int. J. mod. phys. A, 13, 553-568, (1998) · Zbl 0925.81080
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