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The Lie operad is Koszul dual to the commutative operad. In some sense, the data of a formal group is an "elaboration" of the data of a Lie algebra. Is there some corresponding "elaboration" of the data of an $E_\infty$-algebra which "corresponds" to the notion of a formal group under Koszul duality?

Formal groups / formal group laws are not given as the algebras for an operad, I don't expect their "Koszul dual" to be either, so I'm not even sure what Koszul duality should mean here. So strictly speaking, this question doesn't even make sense. Nonetheless, Koszul duality seems to be a phenomenon which has many manifestations, so perhaps there's some adaptation of the notion for which my question at least makes sense.

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I'm not sure what you mean by "elaboration" since the category of formal groups is equivalent to that of Lie algebras. There is however a way Koszul duality can enter this story (I'm not an expert so take this with a pinch of salt).

Formal groups are (by definition) the same as conilpotent cocommutative bialgebras (which are thus automatically Hopf), and the equivalence with Lie algebras comes from Cartier-Milnor-Moore theorem.

I feel like what you're asking for boils down to reproving this theorem using Koszul duality. Namely, I believe if you apply Koszul duality between coalgebras and algebras to the coproduct of a conilpotent cocommutative coalgebra, you'll get something like an algebra in $E_\infty$ algebra, which should just be the same as an $E_\infty$-algebra (by the "stable" version of Dunn thoerem). Applying now the duality between Com and Lie you'll get a dg-Lie algebra.

So long story short I guess I'm trying to say Lie algebra and formal groups are both Kozsul dual to $E_\infty$-algebras and that the composition of these equivalences one way is the enveloping algebra, and composition the other way is taking the primitives.

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    $\begingroup$ Thanks! When you say that Lie algebras and football groups are equivalent, I assume you mean in characteristic zero, right? $\endgroup$
    – Tim Campion
    Commented Dec 13, 2019 at 14:13
  • $\begingroup$ Right, sorry, positive characteristic is way too scary for my taste. $\endgroup$
    – Adrien
    Commented Dec 13, 2019 at 14:15
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    $\begingroup$ Wow, I just noticed how helpful my phone's autocorrect was in that last comment! $\endgroup$
    – Tim Campion
    Commented Dec 13, 2019 at 17:12
  • $\begingroup$ I'm not familiar with how to regard a formal group as a conilpotent cocommutative Hopf algebra. Where can I read about this? $\endgroup$
    – Tim Campion
    Commented Dec 13, 2019 at 17:15
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    $\begingroup$ Well, in general the category of formal schemes is opposite to that of profinite commutative algebras, which itself is opposite to that of cocommutative coalgebras. I think it's now customary to actually define a formal scheme as the "cospectrum" of a cocommutative coalgebra, and it's connected iff the coalgebra is conilpotent I think. Hence formal group schemes are group objects in there, ie conilpotent cocommutative Hopf algebras. This should work in any characteristic I think ? (continued..) $\endgroup$
    – Adrien
    Commented Dec 14, 2019 at 11:48

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