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

TL;DR

This paper constructs explicit realizations of odd-dimensional quantum Euclidean spaces R_q^N using differential operators on classical R^N, extending previous work from R_q^3 to higher odd dimensions.

Contribution

It provides a general method to realize R_q^N for odd N via differential operators, broadening the understanding of noncommutative geometries.

Findings

Realizations for R_q^5 and R_q^7 are explicitly constructed.

The approach is generalized to all odd N, showing the algebra can be represented on classical function spaces.

The results facilitate analysis of quantum spaces using differential operators.

Abstract

The quantum Euclidean space R_{q}^{N} is a kind of noncommutative space which is obtained from ordinary Euclidean space R^{N} by deformation with parameter q. When N is odd, the structure of this space is similar to R_{q}^{3}. Motivated by realization of R_{q}^{3} by differential operators in R^{3}, we give such realization for R_{q}^{5} and R_{q}^{7} cases and generalize our results to R_{q}^{N} (N odd) in this paper, that is, we show that the algebra of R_{q}^{N} can be realized by differential operators acting on C^{infinite} functions on undeformed space R^{N}.