Let $k$ be a field of characteristic $0$.

There is a functor $U$ from Lie-algebras over $k$ to Hopf algebras over $k$ sending a $k$-Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$, and a functor $P$ from the category of Hopf algebras over $k$ to the category of $k$-Lie algebras sending a Hopf algebra $H$ over $k$ to the set $P(H)$ of its primitive elements, which acquires the structure of a Lie algebra. There is an adjoint relationship where $U \dashv P$.

We may ask, "for which $H$ is the counit $\epsilon_H : U(P(H)) \rightarrow H$ of this adjunction an isomorphism". The Milnor-Moore theorem says that $\epsilon_H$ is an isomorphism when $H$ is graded, co-commutative, and generated by its primitive elements.

But, for my purposes, $H$ is not a graded Hopf algebra, merely co-commutative and generated by its primitive elements. I have only been able to find expositions of the Milnor-Moore theorem which use graded Hopf algebras and graded Lie-algebras, but I am interested in the general case.

For instance, I have seen the paper "On the structure of Hopf algebras", by Milnor and Moore, but this seems to work only in the graded case from what I can see.

Could someone direct me to the most accessible exposition (that they know of) of an ungraded version of the Milnor-Moore theorem?

• Your $\epsilon_H$ is still an isomorphism in your case. If $H$ is generated by its primitive elements, then you can build a filtration on $H$: Let $H^{\leq k} = \sum\limits_{i=0}^k \left(P\left(H\right)\right)^i$. It is not hard to check that this is indeed a bialgebra filtration, so your $H$ becomes a connected filtered bialgebra. Now, all you need is the Cartier-Milnor-Moore theorem for connected filtered bialgebras. This is probably in many places; from what I recall, it follows from Duchamp/Minh/Tollu/Chien/Nghia, arXiv:1302.5391v7 for example. – darij grinberg Aug 7 '18 at 19:47
• Another place where you can find the Milnor-Moore theorem for filtered bialgebras is Chapter II of Bourbaki's Lie groups and Lie algebras, §1, Theorem 1. – abx Aug 7 '18 at 19:56
• In your context, could you consider your $H$ to be a graded Hopf algebra, but concentrated purely in degree $0$? Then perhaps your situation would be a special case of the graded situation. – Christopher Drupieski Aug 20 '18 at 13:06

• Sweedler's book: Hopf algebras, Theorem 8.1.5, p 176 (with no particular assumptions on the field) and section 13.1, p. 279, where the version for fields of characteristic $0$ is stated as an exercise.