Representation Functions d_mk^j of U[sl_q(2)] as Wave Functions of ‘Quantum Symmetric Tops’ and Relationship to Braiding Matrices

Abstract

Quantum d-functions d_mk^j, which constitute the (2j+1)-dimensional representation matrix of the quantum group U[sl_q(2)], are investigated to specify them as wave functions of ‘quantum symmetric tops’ in the noncommutative space. It is shown that the d-functions are solutions to the equation RT^j′ T^j″=T^j″ T^j′ R, known in the quantum inverse scattering method, where R is the (2j′+1)×(2j″+1) braiding matrix of U[sl_q(2)]. Quantum d-functions fulfill also Zamolodchikov-Zamolodchikov equation, which affords a new kind of braiding matrix that expresses scattering of a couple of quantum symmetric tops. Explicit forms of quantum d-functions and several symmetry relations are obtained for them. Differential operations are given which describe space- and body-referred angular momentum operators of the quantum symmetric top. Description of quantum d-functions in terms of creation and annihilation operators is also discussed.