NONSTANDARD q-DEFORMATION OF THE EUCLIDEAN LIE ALGEBRA AND ITS REPRESENTATIONS*)

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Abstract

A nonstandard q-deformed Euclidean algebra Uq(ison), based on the definition of the twisted q-deformed algebra Uq(ison) (different from the Drinfeld–Jimbo algebra Uq(son)), is defined. Infinite dimensional representations ℛ of Uq(ison) are described. Explicit formulas for operators of these representations in the orthonormal basis are given. The spectra of the operators ℛ(Tn) corresponding to a q-analogue of the infinitesimal operator of shifts along the n-th axis are described. Contrary to the case of the classical Euclidean Lie algebra ison, these spectra are discrete and spectral points have one point of accumulation.

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