Abstract
We show that the bicrossproduct model C[SU ∗2 ] ▶ ◁ U (su2) quantum Poincaré group in 2+1 dimensions acting on the quantum spacetime [x i , t] = ıλx i is related by a Drinfeld and module-algebra twist to the quantum double U (su2)⊲<C[SU2] acting on the quantum spacetime [x μ , x ν ] = ıλϵ μνρ x ρ . We obtain this twist by taking a scaling limit as q → 1 of the q-deformed version of the above, where it corresponds to a previous theory of q-deformed Wick rotation from q-Euclidean to q-Minkowski space. We also recover the twist result at the Lie bialgebra level.
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References
G. ’t Hooft, Quantization of point particles in (2 + 1)-dimensional gravity and space-time discreteness, Class. Quant. Grav. 13 (1996) 1023 [gr-qc/9601014] [INSPIRE].
E. Batista and S. Majid, Noncommutative geometry of angular momentum space U (su 2), J. Math. Phys. 44 (2003) 107 [hep-th/0205128] [INSPIRE].
L. Freidel and S. Majid, Noncommutative harmonic analysis, sampling theory and the Duflo map in 2 + 1 quantum gravity, Class. Quant. Grav. 25 (2008) 045006 [hep-th/0601004] [INSPIRE].
L. Freidel and E.R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and effective field theory, Class. Quant. Grav. 23 (2006) 2021 [hep-th/0502106] [INSPIRE].
S. Majid and H. Ruegg, Bicrossproduct structure of κ-Poincaré group and noncommutative geometry, Phys. Lett. B 334 (1994) 348 [hep-th/9405107] [INSPIRE].
G. Amelino-Camelia and S. Majid, Waves on noncommutative space-time and gamma-ray bursts, Int. J. Mod. Phys. A 15 (2000) 4301 [hep-th/9907110] [INSPIRE].
J. Lukierski, H. Ruegg, A. Nowicki and V.N. Tolstoi, q-deformation of Poincaré algebra, Phys. Lett. B 264 (1991) 331 [INSPIRE].
S. Majid and B.J. Schroers, q-deformation and semidualisation in 3d quantum gravity, J. Phys. A 42 (2009) 425402 [arXiv:0806.2587] [INSPIRE].
A.Yu. Alekseev, H. Grosse and V. Schomerus, Combinatorial quantization of the Hamiltonian Chern-Simons theory, Commun. Math. Phys. 172 (1995) 317 [hep-th/9403066] [INSPIRE].
S. Major and L. Smolin, Quantum deformation of quantum gravity, Nucl. Phys. B 473 (1996) 267 [gr-qc/9512020] [INSPIRE].
S. Majid, q-Euclidean space and quantum group Wick rotation by twisting, J. Math. Phys. 35 (1994) 5025 [hep-th/9401112] [INSPIRE].
S. Majid, Foundations of quantum group theory, Cambridge University Press, Cambridge U.K., (1995) [INSPIRE].
E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, The three-dimensional Euclidean quantum group E(3) q and its R-matrix, J. Math. Phys. 32 (1991) 1159 [INSPIRE].
S. Majid and R. Oeckl, Twisting of quantum differentials and the Planck scale Hopf algebra, Commun. Math. Phys. 205 (1999) 617 [math/9811054] [INSPIRE].
E. Beggs and S. Majid, Nonassociative Riemannian geometry by twisting, J. Phys. Conf. Ser. 254 (2010) 012002 [arXiv:0912.1553] [INSPIRE].
A. Sitarz, Noncommutative differential calculus on the κ-Minkowski space, Phys. Lett. B 349 (1995) 42 [hep-th/9409014] [INSPIRE].
S. Majid, Hopf algebras for physics at the Planck scale, Class. Quant. Grav. 5 (1988) 1587 [INSPIRE].
V.G. Drinfeld, Quantum groups, in Proc. ICM, Berkeley U.S.A., (1986) [J. Sov. Math. 41 (1988) 898] [Zap. Nauchn. Semin. 155 (1986) 18] [INSPIRE].
P.K. Osei and B.J. Schroers, On the semiduals of local isometry groups in 3d gravity, J. Math. Phys. 53 (2012) 073510 [arXiv:1109.4086] [INSPIRE].
P.K. Osei and B.J. Schroers, Classical r-matrices via semidualisation, J. Math. Phys. 54 (2013) 101702 [arXiv:1307.6485] [INSPIRE].
S. Majid, q-fuzzy spheres and quantum differentials on B q [SU 2] and U q (su 2), Lett. Math. Phys. 98 (2011) 167 [arXiv:0812.4942] [INSPIRE].
D. Gurevich and S. Majid, Braided groups of Hopf algebras obtained by twisting, Pacific J. Math. 162 (1994) 27.
V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge U.K., (1994) [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
E. Witten, (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
V.V. Fock and A.A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and r-matrix, ITEP preprint, Russia, (1992), pg. 72 [Am. Math. Soc. Transl. 191 (1999) 67] [math/9802054] [INSPIRE].
C. Meusburger and B.J. Schroers, Generalised Chern-Simons actions for 3d gravity and κ-Poincaré symmetry, Nucl. Phys. B 806 (2009) 462 [arXiv:0805.3318] [INSPIRE].
S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations, Pacific J. Math. 141 (1990) 311.
S. Majid, Examples of braided groups and braided matrices, J. Math. Phys. 32 (1991) 3246 [INSPIRE].
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ArXiv ePrint: 1708.07999
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Majid, S., Osei, P.K. Quasitriangular structure and twisting of the 3D bicrossproduct model. J. High Energ. Phys. 2018, 147 (2018). https://doi.org/10.1007/JHEP01(2018)147
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DOI: https://doi.org/10.1007/JHEP01(2018)147