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?

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    $\begingroup$ 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. $\endgroup$ – darij grinberg Aug 7 '18 at 19:47
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    $\begingroup$ 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. $\endgroup$ – abx Aug 7 '18 at 19:56
  • $\begingroup$ 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. $\endgroup$ – Christopher Drupieski Aug 20 '18 at 13:06

The "ungraded" version of the theorem -which is actually the version for the Hopf algebras- can be found in most of the classical references on the subject, although its statement and proof appears scattered among paragraphs or several different sections. For example see:

  • 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.
  • Montgomery's book: Hopf algebras and their actions on rings, Corollary 5.6.4, (3), p.78 and Theorem 5.6.5, p. 79 (for zero characteristic),
  • Abe's book: Hopf algebras, in ch. 2, sect. 5.2, p. 110, theorem 2.5.3, deals with the special case of the irreducible, cocommutative hopf algebras over zero characteristic (i am not sure if the pointed case is contained in that book).

Finally, it is worth taking a look at: A primer of Hopf algebras, sect. 3.8, theorem 3.8.2, p.45 for a more modern presentation of the topic.

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