$q$-deformed Euclidean algebras and their representations

A new $q$-deformed Euclidean algebra $U_q(\operatorname {iso}_n)$, based on the definition of the algebra $U_q(\operatorname {so}_n)$ different from the Drinfeld–Jimbo definition, is given. Infinite dimensional representations $T_a$ of this algebra, characterized by one complex number, is described. Explicit formulas for operators of these representations in an orthonormal basis are derived. The spectrum of the operator $T_a(I_n)$ corresponding to a $q$-analogue of the infinitesimal operator of shifts along the $n$-th axis is given. Contrary to the case of the classical Euclidean algebra $\operatorname {iso}_n$, this spectrum is discrete and spectrum points have one point of accumulation.