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We briefly report on our result that, if there exists a realization of a Hopfalgebra $H$ in a $H$-module algebra $A$, then their cross-product is equal tothe product of $A$ itself with a subalgebra isomorphic to $H$ and commutingwith $A$. We illustrate its application to the Euclidean quantum groups in$N\ge 3$ dimensions.Comment: Latex file, 8 pages. Talk given at the 2nd International Symposium on  "Quantum Theory and Symmetries", July 2001, Krakow, Polan

DECOUPLING OF TRANSLATIONS FROM HOMOGENEOUS TRANSFORMATIONS IN INHOMOGENEOUS QUANTUM GROUPS