TITLE: Recovering Hopf Algebra from Group-Like Elements
QUESTION [6 upvotes]: Given the natural coalgebra structure on a group algebra $kG$, one can recover the group by taking the set of group-like elements of the coalgebra $kG$.
When can you go the other way? In particular, given a Hopf algebra $H$, under what conditions can one recover the structure of $H$ from it's group of group-like elements?
I'm also curious as to how the answer differs if $H$ is finitely generated versus finite dimensional.
Thanks!

REPLY [2 votes]: Regarding the question: 

In particular, given a Hopf algebra $H$, under what conditions can one recover the structure of $H$ from it's group of group-like elements?

the simplest case in which this happens corresponds to f.d. cocommutative hopf algebras. In particular, it can be shown that:

If $H$ is a finite dimensional, cocommutative hopf algebra, over an algebraically closed field of characteristic zero, then:
  $$
H\cong kG(H)
$$
  where $G(H)$ is the set of the grouplikes of $H$, which is a group, and $kG(H)$ is its group algebra. 

This is actually a consequence of the Cartier-Konstant-Milnor-Moore theorem (when applied to the finite dimensional case).  
Another interesting situation arises in the case of the commutative hopf algebras: 

If $H$ is a finite dimensional, commutative hopf algebra, over an algebraically closed field of characteristic zero, then:
  $$
H\cong \big(kG(H^*)\big)^*
$$
  where $H^*$ is the dual hopf algebra of $H$, $G(H^*)$ is the set of the grouplikes of $H^*$, $kG(H^*)$ its group hopf algebra and $\big(kG(H^*)\big)^*$ the dual hopf algebra of the group hopf algebra $kG(H^*)$. 

A special case combining both commutativity and cocommutativity is discussed in Reference on correspondence between commutative Hopf Algebras and Groups. Maybe that could also be of some interest for the OP.