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Bicrossproduct structure of $$\kappa$$-Poincaré group and non-commutative geometry. (English) Zbl 1112.81328
Summary: We show that the $$\kappa$$-deformed Poincaré quantum algebra proposed for elementary particle physics has the structure of a Hopf algebra bicrossproduct $$U(\text{so}(1,3))\vartriangleright\!\blacktriangleleft T$$. The algebra is a semidirect product of the classical Lorentz group $$\text{so}(1,3)$$ acting in a deformed way on the momentum sector $$T$$. The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the $$\kappa$$-Poincaré algebra acts covariantly on a $$\kappa$$-Minkowski space, which we introduce. It turns out necessarily to be deformed and noncommutative. We also connect this algebra with a previous approach to Planck scale physics.

##### MSC:
 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 58B34 Noncommutative geometry (à la Connes)
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##### References:
 [1] Lukierski, J.; Nowicki, A.; Ruegg, H.; Tolstoy, V.N., Q-deformation of Poincaré algebra, Phys. lett. B, 268, 331-338, (1991) [2] Lukierski, J.; Nowicki, A.; Ruegg, H., New quantum Poincaré algebra and κ-deformed field theory, Phys. lett. B, 293, 344-352, (1992) · Zbl 0834.17022 [3] Nowicki, A.; Sorace, E.; Tarlini, M., The quantum deformed Dirac equation from the $$κ- Poincaré$$ algebra, Phys. lett. B, 302, 419-422, (1993) [4] Lukierski, J.; Ruegg, H.; Rühl, W., From $$κ- Poincaré$$ algebra to κ-Lorentz quasigroup. A deformation of relativistic symmetry, Phys. lett. B, 313, 357-366, (1993) [5] Biedenharn, L.C.; Mueller, B.; Tarlini, M., The Dirac-Coulomb problem for the $$κ- Poincaré$$ quantum group, Phys. lett. B, 318, 613-616, (1993) [6] Domokos, G.; Kovesi-Domokos, S., Astrophysical limit on the deformation of the Poincaré group, (1993), Preprint JHK-TIPAC-920027/Rev · Zbl 0384.17001 [7] Ruegg, H., Q-deformation of semisimple and nonsemisimple Lie algebras, (), 45-81 · Zbl 0832.17011 [8] Celeghini, E.; Giachetti, E.; Sorace, R.; Tarlini, M., Three-dimensional quantum groups from contractions of SU(2)q, J. math. phys., 31, 2548-2551, (1990) · Zbl 0725.17020 [9] Majid, S.; Majid, S., Hopf algebras for physics at the Planck scale, J. classical and quantum gravity, Phd thesis, 5, 1587-1606, (1988), Harvard · Zbl 0672.16009 [10] Majid, S., Principle of representation-theoretic self-duality, Phys. essays, 4, 3, 395-405, (1991) [11] Majid, S.; Majid, S., Physics for algebraists: non-cummutative and non-cocommutative Hopf akgebras by a bicrossproduct construction, J. algebra, Phd thesis, 130, 17-64, (1988), Harvard · Zbl 0694.16008 [12] Majid, S.; Majid, S., Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pac. J. math., Phd thesis, 141, 311-332, (1988), Harvard · Zbl 0735.17017 [13] Singer, W., Extension theory for connected Hopf algebras, J. alg., 21, 1-16, (1972) · Zbl 0269.16011 [14] H. Ruegg and V.N. Tolstoy, Representation theory of the quantized Poincaré algebra. Tensor operators and their applications to one-particle systems, Preprint UGVA-DPT 1993/07-828. [15] Majid, S.; Majid, S., Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and the classical Yang-Baxter equations, J. funct. analysis, Phd thesis, 95, 291-319, (1988), Harvard · Zbl 0741.46033 [16] Drinfeld, V.G., Quantum groups, (), 798-820 [17] Majid, S., On q-regularization, Int. J. mod. phys. A, 5, 24, 4689-4696, (1990) · Zbl 0745.17022 [18] Zakrzewski, S., Quantum Poincaré group related to the $$κ- Poincaré$$ algebra, J. phys. A, 27, 2075-2082, (1994) · Zbl 0834.17024 [19] Zaugg, Ph., The quantum two dimensional Poincaré group from quantum group contraction, (1994), Preprint MIT-CTP-2294 [20] Mackey, G.W., Induced representations, (1968), Benjamin New York · Zbl 0174.28101 [21] Doebner, H.D.; Tolar, J., Quantum mechanics on homogeneous spaces, J. math. phys., 16, 975-984, (1975) · Zbl 0337.58005
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