Construction of a central extension of a Lie group from its class of symplectic cohomology

Bargmann’s group is a central extension of Galilei group motivated by quantum-theoretical considerations. Bargmann’s work suggests that one of the reasons of the failure of naïve attemps to construct actions on quantum wave functions has a cohomologic origin. It is this point, we develop in the context of Lie groups with symplectic actions. Studying the co-adjoint representation of a central extension of a group G$G$, we highlight the link between the extension cocycles and the symplectic cocycles of G$G$. Also, each extension coboundary corresponds to a symplectic coboundary. Finally, we emphasize the condition to be satisfied by the extension cocycle for the class of symplectic cohomology of the extension being null. The method is illustrated by application to Physics.