  We generalize the differential representation of the operators of the
Galilean algebras to include fractional derivatives. As a result a whole new
class of scale invariant Galilean algebras are obtained. The first member of
this class has dynamical index $z=2$ similar to the Schr\"odinger algebra. The
second member of the class has dynamical index $z=3/2$, which happens to be the
dynamical index Kardar-Parisi-Zhang equation.
