  Fuzzy spaces are obtained by quantizing adjoint orbits of compact semi-simple
Lie groups. Fuzzy spheres emerge from quantizing S^2 and are associated with
the group SU(2) in this manner. They are useful for regularizing quantum field
theories and modeling spacetimes by non-commutative manifolds. We show that
fuzzy spaces are Hopf algebras and in fact have more structure than the latter.
They are thus candidates for quantum symmetries. Using their generalized Hopf
algebraic structures, we can also model processes where one fuzzy space splits
into several fuzzy spaces. For example we can discuss the quantum transition
where the fuzzy sphere for angular momentum J splits into fuzzy spheres for
angular momenta K and L.
