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There is supposedly somewhere in Jacob Lurie’s texts (I guess either in Higher Algebra or in Formal Moduli Problems ) a tower of $\infty$-adjunction between $E_n$-algebras and $L_\infty$-algebras/dg-Lie algebras such that for $n = 1$ it sends a Lie algebra to its ordinary universal enveloping algebra.
Sorry for the stupid question, but: can anyone point out to me the document/page/paragraph that has this discussion? Thanks.
have added a small remark in this direction at Poisson n-algebra here
I have created a tentative entry universal enveloping E-n algebra with some pointers to the literature
I do not understand. The universal enveloping of an ordinary associative algebra is a noncommutative algebra, hence the correct thing is to expect an enveloping $A_\infty$-algebra for $L_\infty$-algebra, and by no means an $E_\infty$-algebra. We discussed that at length several years ago, following the paper
My idea was that the extension of the formalism of Durov in our joint 2007 paper where he gives certain formal Lie theory could combine Getzler’s integration with Baranovsky tools and an analogue of Durov’s geometric framework to get an integration theory without restriction to nilpotent case, but rather in formal geometry.
One of the obstacles was to find the Kan complex in enveloping $A_\infty$-algebra which would correspond to the higher group-like elements. This is also related to establishing the theory of Magnus infinity-group and some other missing technicalities. At Magnus infinity-group you can find some of my past comments on the problem.
The ordinary case is reproduced for $n = 1$. $E_1 \simeq A_\infty$.
This does not look relevant. The ordinary case is $A_1$, not $E_1$. How do you embed the category of $A_1$-algebras into the $(\infty,1)$-category of $E_\infty$ algebras ??
No:
A-1 algebras are not an equivalent to associative algebras. Associative algebras are $A_\infty$-algebras that happen to be concentrated in degree 0.
$E_\infty$ plays no role here.
For discussion of how the universal $E_1$-enveloping algebra of a Lie algebra is its ordinary universal enveloping algebra, see the references pointed to at universal enveloping E-n algebra.
I mean assocaitive algebras in chain complexes, that is dg-algebras. For them, the enveloping algebra theory is a straightfoward, standard and literal extension of the usual case, and used for example in Hinich’s work. Still nothing is commutative. Of course, the formal theory makes formal groups equivalent via integration what makes cocommutative complete equivalent, but this is no point. I will look at the references though.
Hm, I didn’t realize a controversy. I found there just to be a misunderstanding. Could you say what you think is controversial?
The page universal enveloping E-n algebra, stubby as it is, is about universal envelopes of $L_\infty$-algebras. In the context of higher algebra one notices that there is a whole tower of such notions, $\mathcal{U}_n$, as the notion of associative algebras is refined to the whole tower of notions of $E_n$-algebras, for $n \in \mathbb{N}$.
See the discussion around the bottom of p. 18 in John Francis’s text.
The proof the for $n = 1$ this $\mathcal{U}_1$ operation on $L_\infty$-algebras applied to an ordinary Lie algebra reproduces the ordinary enveloping algebra is prop. 4.6.1, p. 78 in Owen Gwilliam’s thesis.
All three links in 10 do not work in current version.
Fixed.
Thanks. It is useful. The thesis is so nice.
How close is the twisted de Rham from this work to the twisted de Rham from works of Gorbounov, Malikov, Schechtman, Gaitsgory where the Gaitsgory’s work was in Beilinson-Drinfeld formalism of chiral algebras ?
Is THIS U1 operation different from the original in Markl-Lada?
I guess it’s equivalent, being left adjoint, as they show, to the evident map the other way round. But I haven’t checked in detail.
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