On the cocycle bicrossed product construction
The cocycle bicrossed product construction, which involves a matched pair of groups and a group of 3-cocycles, was proposed for finite groups by G.I. Kac  and is now used for constructing examples of locally compact quantum groups . A corresponding construction of a Lie bialgebra also exists for a matched pair of Lie algebras .
In the case where the groups in the matched pair are Lie groups, we give a relationship between the groups of differentiable cocycles for the matched pair of groups and the matched pair of the corresponding Lie algebras.