R-matrix formulation of the quantum inhomogeneous groups ISOq,r(N) and ISp(q,r)(N)

The quantum commutations RTT = TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametric q-groups of the B-n, C-n, D-n type are found in terms of the R-matrix of B-n+1, C-n+1, D-n+1. A consistent Hopf structure on these inhomogeneous q -groups is constructed by means of a projection from B-n+1, C-n+1, D-n+1. Real forms are discussed; in particular, we obtain the q-groups ISOq,r(n + 1, n - 1), including the quantum Poincare group. The inhomogeneous q-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-free q-Poincare group depending on one real parameter q.