The phase space of a particle on a group manifold can be split into le
ft and right sectors, in close analogy with the chiral sectors in Wess
-Zumino-Witten models. We perform a classical analysis of the sectors,
and geometric quantization in the case of SU(2). The quadratic relati
on, classically identifying SU(2) as the sphere S3, is replaced quantu
m-mechanically by a similar condition on noncommutative operators (''q
uantum sphere''). The resulting quantum exchange algebra of the chiral
group variables is quartic, not quadratic. The fusion of the sectors
leads to a Hilbert space that is subtly different from the one obtaine
d through a more direct (unsplit) quantization.