Geodesic
$q$-deformed Euclidean algebras and their representations
Teoretičeskaâ i matematičeskaâ fizika, Tome 103 (1995) no. 3, pp. 467-475 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{TMF_1995_103_3_a8,
     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},
     year = {1995},
     volume = {103},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1995_103_3_a8/}
}
TY  - JOUR
AU  - V. A. Groza
AU  - I. I. Kachurik
AU  - A. U. Klimyk
TI  - $q$-deformed Euclidean algebras and their representations
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1995
SP  - 467
EP  - 475
VL  - 103
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1995_103_3_a8/
LA  - ru
ID  - TMF_1995_103_3_a8
ER  - 
%0 Journal Article
%A V. A. Groza
%A I. I. Kachurik
%A A. U. Klimyk
%T $q$-deformed Euclidean algebras and their representations
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1995
%P 467-475
%V 103
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1995_103_3_a8/
%G ru
%F TMF_1995_103_3_a8
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/

[1] Gavrilik A.M., Kachurik I.I., Klimyk A.U., Preprint ITP-90-26E, Kiev, 1990 | MR

[2] Gavrilik A.M., Klimyk A.U., Lett. Math. Phys., 21 (1991), 215 | DOI | MR | Zbl

[3] Drinfeld V.G., Sov. Math. Dokl., 32 (1985), 254

[4] Jimbo M., Lett. Math. Phys., 10 (1985), 63 | DOI | MR | Zbl

[5] Gavrilik A.M., Klimyk A.U., J. Math. Phys., 35:7 (1994) | DOI | MR | Zbl

[6] Gavrilik A.M., Teoret. i Matem. Fizika, 95 (1993), 254 | MR

[7] Gasper G., Rahman M., Basic Hypergeometric Functions, Cambridge Univ. Press, 1990 | MR | Zbl

[8] Vilenkin N.Ja., Klimyk A.U., Representation of Lie Groups and Special Functions, vol. 2, Kluwer, Dordrecht, 1993 | MR | Zbl

[9] Vilenkin N.Ja., Klimyk A.U., Representation of Lie Groups and Special Functions, vol. 1, Kluwer, Dordrecht, 1991 | MR | Zbl

[10] Rozenblyum A.V., Rozenblyum L.V., Multivariate Special Functions in the Theory of Group Representations, Universitetskoye, Minsk, 1992 | MR