Representations of the Q-deformed Euclidean Algebra Uq(iso3) and Spectra of their Operators

Representations of the q-deformed Euclidean algebra Uq(iso3), which at q ! 1 gives the universal enveloping algebra U(iso3) of the Lie algebra iso3 of the Euclidean Lie group ISO(3), are studied. Explicit formulas for operators of irreducible -representations defined by two parameters 2 R and s 2 12Z are given. At q ! 1, these representations exhaust all irreducible infinite-dimensional -representations of U(iso3). The spectrum of the operator T,s (I3) corresponding to a q-analogue of the infinitesimal operator of shifts along the third axis is given. Contrary to the case of the classical Euclidean algebra iso3, this spectrum is discrete and has one point of accumulation.