Notes on a particular Weyl Algebra

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.