The galilean group in 2+1 space-times and its central extension

The problem of constructing the central extensions, by the circle group, of the group of Galilean transformations in two spatial dimensions; as well as that of its universal covering group, is solved. Also solved is the problem of the central extension of the corresponding Lie algebra. We find that the Lie algebra has a three parameter family of central extensions, as does the simply-connected group corresponding to the Lie algebra. The Galilean group itself has a two parameter family of central extensions. A corollary of our result is the impossibility of the appearance of non-integer-valued angular momentum for systems possessing Galilean invariance.