CENTRAL EXTENSIONS AND REALIZATIONS OF ONE-DIMENSIONAL GALILEAN SYSTEMS AND QUANTIZATION

The unitary irreducible realizations (representations up to a factor) of the maximal non-trivial central extension of the (1 + 1) Galilei gr oup, (G) over bar(1 + 1), are obtained via the linear unitary irreduci ble representations of its maximal non-trivial central extension, G do uble over bar(1 + 1). As an application we construct the Stratonovich- Weyl correspondence, which allows Moyal quantization of classical syst ems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.