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We construct the Hopf algebras of Uq(iso(N)) and Uq(iso(3,1)) as regular functionals on their function algebras, examine the possible *-involutions on these objects, derive the quantized Euclidean algebras Uq(e(N)), and finally give the irreducible representations of Uq(e(3)) depending on two parameters which fix the value of the Casimir operators.

We prove the embedding of ISO q (3) ֒→ ISU ex √ q (2) and ISO q (2, 1) ֒→ ISL ex q (2, R) as * -algebras and give a Hilbert space representation of ISU ex √ q (2).