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

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

1994-07-29

arxiv.org/abs/hep-th/9407195v2

J.Math.Phys.36:4363-4405,1995

Physics

High Energy Physics

High Energy Physics - Theory

67 pages, 1 figures. Revised version: Format changed, typos amended, some citations added

Scientific paper

10.1063/1.530898

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 $q\rightarrow 1$.

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