  The unitary implementation of a symmetry group $G$ of a classical system in
the corresponding quantum theory entails unavoidable deformations $\TG$ of $G$,
namely, central extensions by the typical phase invariance group U(1). The
appearance of central charges in the corresponding Lie-algebra quantum
commutators, as a consequence of non-trivial responses of the phase of the wave
function under symmetry transformations, lead to a quantum generation of extra
degrees of freedom with regard to the classical counterpart. In particular,
symmetries of the Hall effect, Yang-Mills and conformally invariant classical
field theories are affected when passing to the quantum realm.
