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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 22nd 2013
• (edited Jan 22nd 2013)

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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJan 24th 2013

have added a small remark in this direction at Poisson n-algebra here

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 24th 2013

I have created a tentative entry universal enveloping E-n algebra with some pointers to the literature

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

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

• Vladimir Baranovsky, A universal enveloping for L-infinity algebras, arxiv/0706.1396

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

The ordinary case is reproduced for $n = 1$. $E_1 \simeq A_\infty$.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

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 ??

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

No:

1. A-1 algebras are not an equivalent to associative algebras. Associative algebras are $A_\infty$-algebras that happen to be concentrated in degree 0.

2. $E_\infty$ 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.

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

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.

• CommentRowNumber9.
• CommentAuthorjim_stasheff
• CommentTimeJan 25th 2013
Given the controversy above, seems to me U(L) for Linfty L deserves a page of its own - or is it there and I've overlooked it? Also why reference to Baronovsky only since his references include 3 earlier works - though he doesn't indicate how his improves on them.
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJan 25th 2013
• (edited Jan 25th 2013)

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.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeJan 25th 2013
• (edited Jan 25th 2013)

All three links in 10 do not work in current version.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJan 25th 2013
• (edited Jan 25th 2013)

Fixed.

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeJan 25th 2013
• (edited Jan 25th 2013)

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 ?

• CommentRowNumber14.
• CommentAuthorjim_stasheff
• CommentTimeJan 27th 2013
@The proof the for n=1 this U1 operation on L∞-algebras applied to an ordinary Lie algebra reproduces the ordinary enveloping algebra is prop. 4.6.1, p. 78 in Owen Gwilliam’s thesis.

Is THIS U1 operation different from the original in Markl-Lada?
• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeFeb 6th 2013

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.