DIFFERENTIAL-CALCULUS ON ISOQ(N), QUANTUM POINCARE ALGEBRA AND Q-GRAVITY

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 ISOq(N) as a consistent projection of the (bicovariant) q-algebra SOq(N + 2). This projection works for particular multiparametric deformations of SO(N + 2), the so-called ''minimal'' deformations. The case of ISOq(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 Poincare 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-Poincare' algebra: the lagrangian generalizes the usual Einstein-Cartan lagrangian.