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Differential calculus on \(ISO_ q(N)\), quantum Poincaré algebra and \(q\)-gravity. (English) Zbl 0848.58004
Summary: We present a general method to deform the inhomogeneous algebras of the \(B_n\), \(C_n\), \(D_n\) type, and find the corresponding bicovariant differential calculus. The method is based on a projection from \(B_{n + 1}\), \(C_{n + 1}\), \(D_{n + 1}\). For example we obtain the (bicovariant) inhomogeneous \(q\)-algebra \(ISO_q (N)\) as a consistent projection of the (bicovariant) \(q\)-algebra \(SO_q (N + 2)\). This projection works for particular multiparametric deformations of \(SO (N + 2)\), the so-called “minimal” deformations. The case of \(ISO_q (4)\) is studied in detail: a real form corresponding to a Lorentz signature exists only for one of the minimal deformations, depending on one parameter \(q\). The quantum Poincaré Lie algebra is given explicitly: it has 10 generators (no dilatations) and contains the classical Lorentz algebra. Only the commutation relations involving the momenta depend on \(q\). Finally, we discuss a \(q\)-deformation of gravity based on the “gauging” of this \(q\)-Poincaré algebra: the Lagrangian generalizes the usual Einstein-Cartan Lagrangian.

MSC:
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
83C45 Quantization of the gravitational field
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