The Euclidean Hopf algebra U q(e N) and its fundamental Hilbert-space representations

Abstract

The Euclidean Hopf algebra Uq(eN) dual of Fun(RNq■SOq−1(N)) is constructed by realizing it as a subalgebra of the differential algebra Diff(RNq) on the quantum Euclidean space RNq; in fact, the previous realization [G. Fiore, Commun. Math. Phys. 169, 475–500 (1995)] of Uq−1(so(N)) is extended within Diff(RNq) 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 RNq, going to distributions in the limit q → 1.