UNITARY QUANTUM GROUPS, QUANTUM PROJECTIVE SPACES AND Q-OSCILLATORS

Abstract

We discuss the parametrization of quantum groups in terms of independent operators. We find that this consideration leads to the parametrization of SU(q)(2) in terms of a q-oscillator plus a commuting phase. The commuting phase is naturally identified with the subgroup U(1)and the remaining coset SU(q)(2)/U(1)=CP(q)(1) consists of a q-oscillator. For unitary quantum groups SU(q)(n), the analogous construction results in the quantum projective space SU(q)(n + 1)/U(q)(n)=CP(q)(n) being identified with the n-dimensional q-oscillator. This yields a nonlinear action of the quantum group SU(q)(n + 1) on the n-dimensional q-oscillator.