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At Hausdorff series I have written some general facts about Magnus algebras and Magnus group. Many related entries need improvement, like Mal’cev completion. The abstract Hausdorff series is about the exponential map in certain completion of the free associative algebra which contains free Lie algebra on the same symbols: the exponentials send Lie elements to invertible elements and all invertibles (elements of the Magnus group) are in the image. If we were dealing with locally nilpotent Lie algebras, instead of free then we do not need to go into completion and do everything within the enveloping algebra which is a Hopf algebra and the exponential there sends primitive elements to group like and all group likes are in the image.
Now what is the infinite-categorical analogue of that picture ? Baranovsky has a version
(arxiv) of the enveloping algebra of an $L_\infty$-algebra $A$ which is of right size (there is a homotopically correct (up to quasi-isomorphism in fact) answer which is too big, namely the cobar construction $\Omega CE(A)$ of the Cartan-Chevalley-Eilenberg coalgebra of $A$, and Baranovsky’s paper aimed at correcting this size issue by producing a smaller version of the right size in the sense of PBW like counting).
On the other hand, Getzler has a hierarchy what he calls “generalized Hausdorff formulas” (or higher Hausdorff formulas) in his Lie integration paper in Annals:
Getzler works with nilpotent case only. I would like to find the group like part and the primitive part both within the infinity version of enveloping and see the role of exponential map. The group like part of enveloping infinity algebra should span a Kan complex. In Baranovsky, the universal enveloping is an A-infinity algebra with coalgebra structure which is strict and the two structures are strictly compatible in my memory. But I do not know how to extract a group-like part from it. What one does for such infinity-bialgebra to extract a Kan complex ?
Next, for non-nilpotent version, what can one say about a completion. Is there an infinity analogue of Magnus algebra and “inside” constructible a Magnus Kan complex ?
For me these questions are very central to the formal infinity-Lie theory.
Getzler’s paper as I said above was about integrating the nilpotent $L_\infty$-algebras. In any case, there should be an infinity analogue of an integration of finite-dimensional Lie algebras to formal Lie group. One interesting variant is due Durov in chapter 7 of
where he considers functors on certain slice category (to work overa base ring) of category of pairs (commutative ring, nilpotent ideal). In infinity setup this has an analogue: commutative $A_\infty$-algebra, nilpotent ideal. Then for a given Lie algebra, he defines several functors on the to enveloping algebra of the Lie algebra restricted to variable basis pairs; here isolating the primitive and group-like part is very important; due nilpotency of the ideal the exponential map is well defined for every single basis pair; while there is no nilpotency needed for the starting Lie algebras. In this setup quite rich elements of formal Lie theory are developed.
It was my idea that a similar thing should work in infinity setup. I discussed this with a number of people, like Urs and, later, Atabey Kaygun in 2008, but defining the group-like part, that is the Kan complex part of the universal enveloping Hopf-infty algebra (reminder: it is a coalgebra, with comultiplication beijng strictly compatible with $A_\infty$-algebra structure) was not solved. There were some interesting ideas, but this basic question unanswered. Maybe perspective which I described in the post 1 above on Magnus group etc. may ring the bell to somebody in $n$Lab now ?
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