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 group, (G) over bar(1 + 1), are obtained via the linear unitary irreducible representations of its maximal non-trivial central extension, G double over bar(1 + 1). As an application we construct the Stratonovich-Weyl correspondence, which allows Moyal quantization of classical systems, for two cases of great physical interest: a system in a external variable force field and a variable-mass system.