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added to Lie algebra a brief paragraph general abstract perspective to go along with this MO reply
I was also looking at Qiaochu’s answer in that MO thread, which I found very nice. Turning to the nLab, I found related material under differential graded coalgebra and differential graded Hopf algebra, which looks nice, but I wanted to understand some of this at a simpler level first.
Is there an adjunction $U \dashv P$ where $P$ takes a cocommutative Hopf algebra to the space of primitives (which I guess is a LIe algebra), and $U$ takes a Lie algebra to its universal enveloping algebra, viewed as a Hopf algebra?
The answer to my last question is a straightforward ’yes’. It’s amazing that such an easy and basic result isn’t seen in the accounts I’ve looked at.
I’ve added some more material to primitive elements. (The previous version was somewhat flawed because we really need to work in the context of unital coalgebras. It’s a little surprising how many authors forget to mention this!)
I think I had seen something like that in some of the rational homotopy theory stuff and further back some related ideas in Quillen’s rational homotopy theory paper, but was unable to check so did not mention it. Certainly something like that is mentioned in Tanré’s lecture notes I seem to remember.
@Todd #3: The remarks before Theorem 22.3.1 in More Concise Algebraic Topology come tantalizingly close:
A quick calculation shows that the $R$-module $P A$ of primitive elemnts of a Hopf algebra $A$ is a Lie subalgebra. The universal property of $U(P A)$ thus gives a natural map of Hopf algebras $g:U(P A)\to A$…. Let $\mathcal{L}$ and $\mathcal{P H}$ denote the categories of Lie algebras and primitive Hopf algebras over $R$. We have functors $U:\mathcal{L}\to \mathcal{P H}$ and $P:\mathcal{P H} \to \mathcal{L}$, a natural inclusion $L\subset P(U L)$, and a natural epimorphism $g:U(P A) \to A$… This much would be true over any commutative ring $R$, but when $R$ is a field of characteristic zero we have the following result.
Theorem 22.3.1. The functors $U$ and $P$ are inverse equivalences of categories.
Yes, that’s just it! “Tantalizingly close.” (Admittedly, this is the closest I’ve seen yet, so thanks for mentioning! To me it makes sense just to come out and state the adjunction, which holds in maximal generality, and then make a more refined analysis that examine the unit and counit separately under more specific hypotheses.)
Yeah, to me too.
The commutative triangle of functors involving two adjunctions, one mentioned above, is stated at the beginning of the paper
quoted at Loday-Pirashvili category.
Nice – thank you Zoran!
touched the section Definition – Internal to a general linear category
(added formatting, missing words, hyperlinks)
Will be splitting this off now as a stand-alone entry Lie algebra object.
touched the section Definition – Internal to a general linear category
(added formatting, missing words, hyperlinks)
Will be splitting this off now as a stand-alone entry Lie algebra object.
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