The Euclidean Hopf algebra $U_q(e^N)$ and its fundamental Hilbert space representations

TL;DR

This paper constructs the Euclidean Hopf algebra $U_q(e^N)$ as a subalgebra of the differential algebra on quantum Euclidean space, revealing its fundamental Hilbert space representations as lattice-regularized highest weight types.

Contribution

It introduces a realization of $U_q(e^N)$ within the differential algebra on quantum Euclidean space, extending previous work and analyzing its fundamental Hilbert space representations.

Findings

Fundamental representations are of highest weight type.

Representations are lattice-regularized versions of classical ones.

Basis vectors can be realized as normalizable functions on quantum space.

Abstract

We construct the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross SO_{q^{-1}}(N))$ by realizing it as a subalgebra of the differential algebra $\DFR$ on the quantum Euclidean space $\rn_q^N$; in fact, we extend our previous realization \cite{fio4} of $U_{q^{-1}}(so(N))$ within $\DFR$ through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of $U_q(e^N)$ turn out to be of highest weight type and rather simple `` lattice-regularized '' versions of the classical ones. The vectors of a basis of the singlet (i.e. zero-spin) irrep can be realized as normalizable functions on $\rn_q^N$, going to distributions in the limit $q\rightarrow 1$.