  The phase space of a particle on a group manifold can be split in left and
right sectors, in close analogy with the chiral sectors in Wess Zumino Witten
models. We perform a classical analysis of the sectors, and the geometric
quantization in the case of $SU(2)$. The quadratic relation, classically
identifying $SU(2)$ as the sphere $S^3$, is replaced quantum mechanically by a
similar condition on non-commutative operators ('quantum 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 obtained by a more direct (un--split)
quantization.
