Realization of $U_q(so(N))$ within the differntial algebra on ${\bf R}_q^N$

TL;DR

This paper constructs a realization of the quantum algebra $U_q(so(N))$ as differential operators on the quantum Euclidean space, providing explicit representations and decompositions into irreducible components.

Contribution

It explicitly realizes $U_q(so(N))$ within a differential algebra framework on ${f R}_q^N$, including the construction of irreducible vector representations.

Findings

Explicit realization of $U_q(so(N))$ as differential operators

Decomposition of function space into irreducible representations

Construction of highest weight representations

Abstract

We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are suitable q-deformed analogs of the angular momentum components on ordinary ${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$ splits into a direct sum of irreducible vector representations of $U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight representations.