I'm reading Akhil Mathew's [blog post](https://amathew.wordpress.com/2012/06/23/formal-lie-theory-in-characteristic-zero/) on Formal Lie Theory in Characteristic Zero.


Let $H$ be cocommutative Hopf algebra over a field $k$. We can form $\mathfrak{g}$, the Lie algebra over $k$ consisting of the primitive elements of $H$. There is a canonical map $\phi : U(\mathfrak{g}) \rightarrow H$ from the universal enveloping algebra of $\mathfrak{g}$ to $H$. In fact, the functor $P : \textbf{Hopf} \rightarrow \textbf{Lie}$ sending a Hopf algebra over $k$ to its subspace of primitive elements (which forms a Lie algebra over $k$) is right adjoint to the universal enveloping algebra functor $U : \textbf{Lie} \rightarrow \textbf{Hopf}$, under certain finiteness conditions on both categories.


I am wondering, 


(a) Under what conditions is $\phi$ (as above) surjective? Of course, if $H$ is generated by its primitive elements then this is true.


(b) When will $\text{im}(\phi)$ contain the group-like elements of $H$?



When is this map of Hopf algebras Surjective?
Some thoughts, regarding question (a): 


In the case of a [pointed](http://library.msri.org/books/Book43/files/andrus.pdf), cocommutative hopf algebra $H$ over a field $k$ of characteristic $0$, by the Cartier-Konstant-Milnor-Moore theorem (see: [Classification of quasitriangular Hopf algebras](https://mathoverflow.net/questions/255767/classification-of-quasitriangular-hopf-algebras)) we have that 
$$
H\cong U\big(P(H)\big)\sharp kG(H)
$$
where $P(H)$ is the lie algebra of the primitives, $U(.)$ its UEA and $G(H)$ the group of the group-likes. 
So, it turns, that a general condition for $\phi$ to be surjective, is the set of grouplikes to be trivial, in which case $H$ is generated by its primitives and $H\cong U\big(P(H)\big)$.  

(this includes as a special case the cocommutative hopf algebras over algebraically closed fields, since these are pointed). 


A similar result -again refering to a case with trivial grouplikes- is the situation of an [irreducible](http://www.ams.org/journals/tran/1972-163-00/S0002-9947-1972-0292875-1/S0002-9947-1972-0292875-1.pdf), cocommutative hopf algebra $H$ over a field $k$ of characteristic zero: then we have the hopf algebra isomorphism 
$$
H\cong U\big(P(H\big))
$$
If the above examples are of interest for your purposes, maybe the article: [Hopf algebras with one groupl-like element](https://www.ams.org/journals/tran/1967-127-03/S0002-9947-1967-0210748-5/S0002-9947-1967-0210748-5.pdf) and [these notes](http://user.math.uzh.ch/stufler/lec/2018hopf/hopf.pdf) (especially sect. 9) will be useful. 


**Edit:** Similar topics are discussed in most of the classical books on the subject (Sweedler's book, Abe's book, Montgomery's etc), but one of the most valuable references -imo- still remains the seminal paper of Milnor and Moore: [On the structure of Hopf algebras](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf) (especially the discussion on p. 239).  

(however it is important to keep in mind that in the above paper the authors define hopf algebras to be what is nowadays called "$\mathbb{Z}$-graded hopf algebras" and generally they use the terminology in a slightly different way than what is understood to be "mainstream" today). 


