Summary: The starting point is the nonsemisimple, inhomogeneous Lie algebra \(U_ n\times I_{2n}\) [denoted also as \(IU(n)]\), where \(I_{2n}\) represents an Abelian subalgebra in semidirect product with the homogeneous part \(U(n)\). This is realized by explicitly giving the matrix elements of the generators on a modified Gelfand-Zetlin basis that allows representations of infinite dimensions. The enveloping algebra is \(q\) lifted by introducing \(q\) brackets in the matrix elements giving \(U_ q(IU(n))\). The deformation of the Abelian structure of \(I_{2n}\) is studied for \(q\neq 1\). Some implications are pointed out. The important invariants are constructed for arbitrary \(n\). The results are compared to the corresponding ones for Jimbo’s construction of \(U_ q(U(n+1))\) on a Gelfand-Zetlin basis. Finally, the related construction of \(U_ q(U(n,1))\) is presented and discussed. Here, \(U_ q(SU(1,1))\), the \(q\)- analog of relativistic motion in a plane, is analyzed in the context of this formalism.