The purpose of this paper is to consider some basic constructions in the
category of compact quantum groups --for example de case of extensions, of
Drinfeld twists, of matched pairs, of extensions, of linked pairs and of
cocycle Singer pairs -- with special emphasis in the finite dimensional
situation. We give conditions, in some cases necessary and sufficient, to
extend to the new objects the original compact structure. We illustrate the
results in the case of matched pairs of groups
