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

The quantum commutationsRTT=TTR and the orthogonal (symplectic) conditions for the inhomogeneous multiparametricq-groups of theB _n ,C _n ,D _n type are found in terms of theR-matrix ofB _n+1 ,C _n+1 ,D _n+1 .A consistent Hopf structure on these inhomogeneousq-groups is constructed by means of a projection fromB _n+1 ,C _n+1 ,D _n+1 .Real forms are discussed; in particular, we obtain theq-groups ISO _q,r (n+1,n–1), including the quantum Poincaré group. The inhomogeneusq-groups do not contain dilatations when the parameters satisfy certain conditions. For example, we find a dilatation-freeq-Poincaré group depending on one real parameterq.