The Euclidean Hopf algebra Uq(eⁿ) and its fundamental Hilbert space representations

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We construct the Euclidean Hopf algebra Uq(eⁿ) dual of Fun(ⁿ SOq⁻¹(N)) by realizing it as a subalgebra of the differential algebra on the quantum Euclidean space ⁿ; in fact, we extend our previous realization fio4 of Uq⁻¹(so(N)) within through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of Uq(eⁿ) 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 ⁿ, going to distributions in the limit q→ 1.