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=2z=2 similar to the Schrödinger algebra. The second member of the class has dynamical index z=3/2z=3/2, which happens to be the dynamical index Kardar–Parisi–Zhang equation.