  By means of the notions of cross product algebras of the theory of quantum
groups, in the context of classical Hopf algebra structures, we deduce some
known structures of Weyl algebras type (as the Drinfeld quantum double, the
restricted Heisenberg double, the generalized Schroedinger representation, and
so on) that may be considered as a non-trivial examples of quantum groups
having physical meaning, starting from a particular example of groupoid
motivated by elementary quantum mechanics.
