QUASI-CONTINUOUS SYMMETRIES OF NON-LIE TYPE

We introduce a smooth mapping of some discrete space-time symmetries i nto quasi-continuous ones. Such transformations are related with q-def ormations of the dilations of the Euclidean space and with the noncomm utative space. We work out two examples of Hamiltonian invariance unde r such symmetries. the Schrodinger equation for a free particle is inv estigated in such a noncommutative plane and a connection with anyonic statistics is found.