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Polynomial extensions of the Weyl $C^*$-algebra

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.