Measuring comodules - their applications

Measuring comodules are defined and shown to provide a useful generalization of the set of maps between modules with a broad range of applications. Three applications are described. Connections on bundles are described in terms of measuring comodules, enabling curvature to be defined under general algebraic circumstances. Loop algebras are realized via a short exact sequence of measuring comodules, with the central extension given by the curvature. Finally dual comodules provide a method of dualizing representations, which, when applied to representations of loop algebras yield positive energy representations, and when applied to representations of totally disconnected groups leads to the smooth dual.