Abstract
Using the general theory of [10], quantum Poincaré groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most cases we solve them and give the classification of quantum Poincaré groups. Each of them corresponds to exactly one quantum Minkowski space. The Poincaré series of these objects are the same as in the classical case. We also classify possibleR-matrices for the fundamental representation of the group.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Chaichian, M., Demichev, A.P.: Quantum Poincaré group, Phys. Lett.B304, 220–224 (1993) Cf. also Schirrmacher, A.: Varieties on quantized spacetime. In: Symmetry methods in physics. A.N. Sissakin et al. (eds.), vol. 2, Moscow: Dubna 1994, pp. 463–470
Dobrev, V.K.: Canonicalq-deformations of noncompact Lie (super-) algebras. J. Phys. A: Math. Gen.26, 1317–1334 (1993)
Kondratowicz, P., Podleś, P.: Irreducible representations of quantumSL q(2) groups at roots of unity. hep-th 9405079
Lukierski, J., Nowicki, A., Ruegg, H.: New quantum Poincaré algebra andk-deformed field theory. Phys. Lett.B293, 344–352 (1992); Zakrzewski, S.: Quantum Poincaré group related to thek-Poincaré algebra. J. Phys. A.: Math. Gen.27, 2075–2082 (1994); Cf. also Lukierski, J., Nowicki, A., Ruegg, H., Tolstoy, V.N.:q-deformation of Poincaré algebra. Phys. Lett.B264, 331–338 (1991)
Majid, S.: Braided momentum in theq-Poincaré group. J. Math. Phys.34, 2045–2058 (1993)
Ogievetsky, O., Schmidke, W.B., Wess, J., Zumino, B.:q-Deformed Poincaré algebra. Commun. Math. Phys.150, 495–518 (1992)
Podleś, P.: Complex quantum groups and their real representations. Publ. RIMS, Kyoto University28, 709–745 (1992)
Podleś, P., Woronowicz, S.L.: Quantum deformation of Lorentz group. Commun. Math. Phys.130, 381–431 (1990)
Podleś, P., Woronowicz, S.L.: Inhomogeneous quantum groups. Submitted to Proceedings of First Caribbean School of Mathematics and Theoretical Physics in Guadeloupe, 1993
Podleś, P., Woronowicz, S.L.: On the structure of inhomogeneous quantum groups. hep-th 9412058, UC Berkeley preprint PAM-631
Schlieker, M., Weich, W., Weixler, R.: Inhomogeneous quantum groups. Z. Phys. C.-Particles and Fields53, 79–82 (1992); Inhomogeneous quantum groups and their universal enveloping algebras. Lett. Math. Phys.27, 217–222 (1993)
Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613–665 (1987)
Woronowicz, S.L.: New quantum deformation ofSL(2, C). Hopf algebra level. Rep. Math. Phys.30(2), 259–269 (1991).
Woronowicz, S.L., Zakrzewski, S.: Quantum deformations of the Lorentz group. The Hopf *-algebra level. Comp. Math.90, 211–243 (1994)
Zakrzewski, S.: Geometric quantization of Poisson groups-diagonal and soft deformations. Contemp. Math.179, 271–285 (1994)
Zakrzewski, S.: Poisson Poincaré groups. Submitted to Proceedings of Winter School of Theoretical Physics, Karpacz 1994, hep-th 9412099 (Some errors are corrected in the last reference); Cf. also Poisson homogeneous spaces. Submitted to Proceedings of Winter School of Theoretical Physics, Karpacz 1994, hep-th 9412101; Poisson structures on the Poincaré group, in preparation
Author information
Authors and Affiliations
Additional information
Communicated by A. Connes
This research was supported in part by NSF grant DMS92-43893 and in part by Polish KBN grant No 2 P301 020 07.
This research was supported by Polish KBN grant No 2 P301 020 07.
Rights and permissions
About this article
Cite this article
Podleś, P., Woronowicz, S.L. On the classification of quantum Poincaré groups. Commun.Math. Phys. 178, 61–82 (1996). https://doi.org/10.1007/BF02104908
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02104908