Exponential map and Lie correspondence within a Hopf algebra setting

Question

The Cartier-Konstant-Milnor-Moore (et al.) theorem for Hopf algebras states that a cocommutative Hopf algebra over $\mathbb{C}$ is isomorphic to a smash product of a universal enveloping algebra of a Lie algebra and a group Hopf algebra. Hence in some sense Hopf algebras manage to interpolate between groups and Lie algebras. As a natural setting for both sides of the Lie correspondence, is there any map known within the realm of Hopf algebras that connects the two?

I have seen examples before where people adjoin the (formal) exponential of a primitive element to an algebra to function as for example a quasitriangular structure or as a way to construct a new algebra (eg. Majid '95 lemma 3.1.3), so I am aware that the exponential of a primitive element forms a grouplike element. I also know that in Lusztig's Introduction to Quantum Groups the pairing between a group algebra on a lie group and the corresponding lie algebra is defined, so there is some sort of duality one can formulate within the language of Hopf algebras. I am looking for any result that instead does something like implement the Lie functor as an endofunctor on a (sub-)category of Hopf algebras.

Answer 1

The Hopf dual is an endofunctor on the category of Hopf algebras that behaves something like the functor you are looking for. It is well explained in Sweelder's book on Hopf algebras.