Fractional Galilean Symmetries

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.