  We introduce higher order (polynomial) extensions of the unique (up to
isomorphisms) non trivial central extension of the Heisenberg algebra. Using
the boson representation of the latter, we construct the corresponding
polynomial analogue of the Weyl $C^*$-algebra and use this result to deduce the
explicit form of the composition law of the associated generalization of the
1-dimensional Heisenberg group. These results are used to calculate the vacuum
characteristic func- tions as well as the moments of the observables in the
Galilei algebra. The continuous extensions of these objects gives a new type of
second quantization which even in the quadratic case is quite different from
the quadratic Fock functor.
