Realization of the N(odd)-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space R_q^N is a kind ofnoncommutative space that is obtained from ordinary Euclidean spaceR^N by deformation with parameter q. When N is odd, thestructure of this space is similar to R_q³. Motivated byrealization of R_q³ by differential operators inR³, we give such realization for R_q⁵ andR_q⁷ cases and generalize our results toR_q^N (N odd) in this paper, that is, we show that thealgebra of R_q^N can be realized by differential operatorsacting on C^∝ functions on undeformed spaceR^N.