The starting point is the nonsemisimple, inhomogeneous Lie algebra U(n) x I2n [denoted also as IU (n)], where I2n 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 I2n is studied for q not-equal 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 Gelfeld-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.
