Differential calculus on ISO_q(N), quantum Poincaré 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 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.