Geodesic
$q$-deformed Euclidean algebras and their representations
Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 3, pp. 467-475

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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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     author = {V. A. Groza and I. I. Kachurik and A. U. Klimyk},
     title = {$q$-deformed {Euclidean} algebras and their representations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {467--475},
     publisher = {mathdoc},
     volume = {103},
     number = {3},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_103_3_a8/}
}
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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/