Cohomology,central extensions,and (dynamical) groups

We analyze in this paper the process of group contraction which allows the transition from the Einstenian quantum dynamics to the Galilean one in terms of the cohomology of the Poincaré and Galilei groups. It is shown that the cohomological constructions on both groups do not commute with the contraction process. As a result, the extension coboundaries of the Poincaré group which lead to extension cocycles of the Galilei group in the nonrelativistic limit are characterized geometrically. Finally, the above results are applied to a quantization procedure based on a group manifold.