Duaity for topological Hopf algebras associated to some discrete and profinite groups

In 2008, Akbarov introduced a framework of holomorphic duality for topological Hopf algebras. Unlike the $C^*$-duality theory, which deals with locally compact groups, Akbarov's theory comes from complex Lie groups. We discuss a simplified version of this framework, which does not involve stereotype spaces and which is based on using strong duals of locally convex spaces. One of our goals is to find applicability limits of this simplified framework. I will briefly explain the duality for connected Lie groups, and then we will turn to another extreme case, namely to discrete groups. We will show that the reflexivity holds for finitely generated and locally finite groups. The second class leads naturally to profinite groups. While they are not Lie groups, it is still possible to define the holomorphic duality and to prove the reflexivity for them.