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Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity

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Abstract

We present a general method to deform the inhomogeneous algebras of theB n,Cn,Dn type, and find the corresponding bicovariant differential calculus. The method is based on a projection fromB n+1,Cn+1,Dn+1. For example we obtain the (bicovariant) inhomogeneousq-algebraISO q(N) as a consistent projection of the (bicovariant)q-algebraSO q(N=2). This projection works for particular multiparametric deformations ofSO(N+2), the so-called “minimal” deformations. The case ofISO q(4) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameterq. The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains theclassical Lorentz algebra. Only the commutation relations involving the momenta depend onq. Finally, we discuss aq-deformation of gravity based on the “gauging” of thisq-Poincaré algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.

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  1. Sezione di Torino, and Dipartimento di Fisica Teorica, Istituto Nazionale di Fisica Nucleare, Via P. Giuria 1, I-10125, Torino, Italy

    Leonardo Castellani

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  1. Leonardo Castellani
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Communicated by A. Connes

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Castellani, L. Differential calculus onISO q(N), quantum Poincaré algebra and q-gravity. Commun.Math. Phys. 171, 383–404 (1995). https://doi.org/10.1007/BF02099276

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  • DOI: https://doi.org/10.1007/BF02099276

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