Geodesic
$q$-deformed Euclidean algebras and their representations
Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 3, pp. 467-475 
V. A. Groza; I. I. Kachurik; A. U. Klimyk. $q$-deformed Euclidean algebras and their representations. Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 3, pp. 467-475. http://geodesic.mathdoc.fr/item/TMF_1995_103_3_a8/
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     title = {$q$-deformed {Euclidean} algebras and their representations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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