In this paper we study certain interrelations between subsets of Hopf algebras H and their duals which stem from various morphisms T defined on H. We define and study T-cocommutative elements. Cocommutative elements, generators of H as an H*-module and invariants under various actions and coactions are all examples. An important role is played by Nakayama automorphisms. These ideas are applied to (co)quasitriangular Hopf algebras, in particular when they are factorizable. Then the Drinfeld element gives rise to a Nakayama automorphism and to a special element in the dual. Explicit calculations are carried out for the Drinfeld double.