THE EUCLIDEAN-HOPF ALGEBRA U-Q(E(N)) AND ITS FUNDAMENTAL HILBERT-SPACE REPRESENTATIONS

The Euclidean Hopf algebra U-q(e(N)) dual of Fun(R(q)(N) X SOq-1(N)) is constructed by realizing it as a subalgebra of the differential algebra Diff(R(q)(N)) on the quantum Euclidean space R(q)(N); in fact, the previous realization [G. Fiore, Commun. Math. Phys. 169, 475-500 (1995)] of U-q-1(so(N)) is extended within Diff(R(q)(N)) 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 father 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 R(q)(N), going to distributions in the limit q --> 1. (C) 1995 American Institute Physics.