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wrote Maurer-Cartan form
the first part is the standard story, but I chose a presentation which I find more insightful than the standard symbol chains as on Wikipedia.
then there is a section on Maurer-Cartan forms on oo-Lie groups and how that reduces to the standard story for ordinary Lie groups.
The detailed statements and proofs of this second part are at Lie infinity-groupoid in the new section The canonical form on a Lie oo-group that is just a Lie group.
In the entry Lie integration, the formula in the middle is overextendeding into floating TOC (one of the reasons I dislike common TOC-s included into each entry of a cricle of entries is exactly bad formatting, especially for printing).
I’m also not a big fan of floating TOCs generally (although I understand why others like them). I wish there was some way, e.g. some setting or option, that would turn them off.
I fixed the formula at Lie integration and in the course of it reworked the entire entry.
I feel that the floating TOCs are crucial for making the nLab navigatable. I think we’d need many more of them. In fact I think that ideally each and every entry came with the TOCs that applied to it.
But I see why it is bad that they display by default, with no way to get rid of them. We should try to put energy into finding some way to make them into something like pulldown menus or the like.
Which reminds me: haven’t heard of Andrew Stacey in a long while…
A similar very-abstract-nonsense™ as for the Maurer-Cartan form on an $\infty$-Lie group gives the vertical form on a $G$-principal $\infty$-bundle. I recorded the relevant diagram here.
I have a proof on paper that this reproduces indeed the ordinary notion of flat vertical forms on ordinary $G$-principal bundles, but no time to type it into the Lab right now…
5 – this is I guess more or less a Lie integrated version of the statement in the dg context – the correspondence at twisting cochain, section Relation to the adjunction bar-cobar between chain maps from C→BA and the twisting cochains, satisfying the Maurer-Cartan by definition.
We should try to put energy into finding some way to make them into something like pulldown menus or the like.
2-3 months ago there was a programmer/volunteer asking what should be nice to be done to improve the software of nlab. So I think the option of choice between expanded and non-expanded floating TOC (that is reduced to a single button somewhere on the top or bottom, but not on the side). In print version, non/expanded should be default I think. Who wants to print just floating TOC can go into the entry of floating TOC itself, not into the page referring to it. I myself almost never use floating TOC on the side, but rather the TOC pages as standalone, like mathematicscontents.
A similar very-abstract-nonsense™
Urs is hopelessly leaving physics for more-and-more-abstract-nonsense™.
this is I guess more or less a Lie integrated version of the statement in the dg context – the correspondence at twisting cochain, section Relation to the adjunction bar-cobar between chain maps from C→BA and the twisting cochains, satisfying the Maurer-Cartan by definition.
Yeah, maybe there is some relation.
2-3 months ago there was a programmer/volunteer asking what should be nice to be done to improve the software of nlab. So I think the option of choice between expanded and non-expanded floating TOC (that is reduced to a single button somewhere on the top or bottom, but not on the side). In print version, non/expanded should be default I think. Who wants to print just floating TOC can go into the entry of floating TOC itself, not into the page referring to it. I myself almost never use floating TOC on the side, but rather the TOC pages as standalone, like mathematicscontents.
One simple hack would be to take all existing floating tocs and replace their content by a single link to a standalone toc.
But there must be something nicer. It’s possible to create scrollbar boxes on our wiki-pages, so it must be possible to have someting like a pulldown menu box, too, I expect.
Urs is hopelessly leaving physics for more-and-more-abstract-nonsense™.
And I think I am trying to understand the differential cohomology of string backgrounds. But it’s true, it’s taking me through quite a detour…
I would much more like to have option than the pull-down menus. Menus are bad for saving, they take width in the browser, they may have more complcaited sourcecode and so on. I would to have option of having no ftoc whatsoever in print versions, and just a link in the normal view. If the browser remembers who am I as a contributor it can remember some flag option as well.
Urs wrote:
Which reminds me: haven’t heard of Andrew Stacey in a long while…
Andrew has been on vacation for a while. He’s coming back on July 19th. Then he’s gonna help me set up a high-tech blog and forum for Azimuth, based in part on the stuff he’s done here!
Dropping in early, I would say that floating TOCs are one of those things that no two people are going to agree on completely, and so it is something where user customisability comes to the fore. Firefox, and probably other modern browsers that aren’t completely broken, has the ability to impose a style on the top of the given one. In firefox, the Stylish extension is one that does this. Using that, one could make the floating TOCs appear wherever one liked, make them a fixed height, make them “drop-down” like menus, … the possibilities, whilst not quite endless, are nonetheless quite large.
The default view should be one that is most useful to someone happening by the nLab for the first time. Whilst the current style is not pretty, nothing is hidden from sight and that’s a good thing. But it wouldn’t be hard to build up a stock of little style tweaks that people who think it worth the (small) effort can install and so customise the look as they like.
Section “gauge transformation” in Maurer-Cartan form, near the bottom, has $Ad_g A$ instead of $g^{-1}A g$ in one of the formulas. Is this an (unusual) convention or a typo ? If the latter, I will change it to $Ad_{g^{-1}} A$.
Wait a second, I am aa bit confused with the synthetic definition. It says that for $x$ infinitesimally close to $y$ in $G$, $y = x \tau(x,y)$ gives element $\tau(x,y)\in G$ which is actually in the Lie algebra. In my view, it should be $id$ plus an element in a Lie algebra.
Feel free to change it.
There is a host of $\mathbb{Z}_2$-choices here. Left or right $G$-actions, $\theta$ left or right invariant, etc.
We had settled on “smooth infinity-group” a good while back. I have made the corresponding terminology changes to Maurer-Cartan form.
But, generally, there should not be any confusion, either way. (In 50 years from now we’ll be dropping all the “$\infty$-“s anyway ;-)
Urs 14: this answers my 12, but not more important 13.
Hi Zoran,
thanks for alerting me of #13. I had not seen that! I guess we overlapped again when I typed #14.
The formula is correct, because this is in terms of the multiplicative description. Instead of the “id plus epsilon” that you are expecting to see we here have “id times epsilon = epsilon”.
But it is easiest to just trust the synthetic logic and let it guide you. That means, we just work as if with finite group elements, and let the formalism take care of making sense of infinitesimals.
So we have two group elements $x$ and $y$ and want to consider their difference. That’s the group element $x^{-1} \cdot y$. If $x$ and $y$ are infinitesimally close, we give that a name: $\theta(x,y)$. That’s it.
To see the relation to the expression $id + \epsilon$ that you are expecting to see, check out in Anders Kock’s book (the second one) the section that is called something like “the exp/log relation” or the like. He explains it in much detail.
Well, I know there is a bijection, but writing $id + \epsilon$ or writing $exp(\epsilon)$ is the same thing for such infinitesimals. In precise notation one needs to take care of all identifications. So if you want to say $log(x^{-1}\cdot y)$, I agree. Group element $\theta$ is not a Lie algebra element (like what the entry says), but its exponential.Thus the entry needs either writing $log$ or writing $id+\epsilon$, it is the same. I agree, and saw at the beginning that I can write $y = x \theta(x,y)$ just I disagree in 13 with the entry that $\theta$ is in the Lie algebra. Instead, $\theta(x,y) = id+\epsilon(x,y)$ where $\epsilon(x,y)$ is in the Lie algebra. That is what I am saying in 13 and still think so. What I see in Kock, agrees with my attitude about this formula (e.g. proof of 4.3.1). Now $\omega_{MC}(y-x) := \epsilon(x,y)$ where $y-x$ is the tangent vector from $x$ to $y$. This is well defined and $\mathfrak{g}$-valued. The entry misleads by stating that $\omega_{MC}(y-x) = \theta(x,y)$ which is not true in my opinion (nor makes sense as this would be a $G$-valued form).
Group element $\theta$ is not a Lie algebra element
Well, the point of the synthetic reasoning is that the difference disappears. The Lie algebra of a group $G$ is the infinitesimal neighbourhood of the neutral element in $G$.
Sometimes
If you let me know explicitly where you find what ambiguous, I’ll be happy to do something about it.
But if you just say it in this generality I can’t do more than say “Sure.”
I understand that there is a bijection. This is not new with the synthetic geometry. In algebraic geometry one has the isomorphism between the underlying spaces of the formal group and of the Lie algebra. But it is not the same, it is just isomorphic (at the level of underlying spaces), and the precise formulas do involve the identification maps which are identification for just part of the structure. For example, the Hopf structure for the enveloping algebra and for the formal group are not isomorphic; there is a difference between the group likes and primitive elements. For nilpotent Lie algebras, where the exponentials are finite, both group likes and primitive elements live in the enveloping algebra, and they are different parts of the enveloping algebra, so we are talking about identifying different subsetws of the same set, what one needs really to be careful with. We have the same in the paper with Durov, where the 4 different tangent functors are in the interplay to take care of the various kinds of neighborhoods.
I’d be happy to see more discussion of the additive picture under the exp/log isomorphism added. All I meant to say is that the present statement in the entry is correct in its sense, and should not be replaced. But it certainly deserves to be expanded on.
in the entry is correct in its sense, and should not be replaced
Are you saying that my proposal (Id + epsilon or exp(epsilon)) is any less correct ? And it agrees with the language of formal groups what is the same thing. Apealing to a “sense” is not necessary if the entities have their unique mathematical identity, but can be used in practice freely, with identity kept in the definition.
No, I am just saying that what it is the entry is correct. When you edit it, please rather expand on what is there, than remove what is there.
I did not propose removing. I proposed writing that $\theta$ is the same as $Id + \epsilon$ and that $\epsilon$ is in the tangent space, literaly, by the definition in Kock. $\theta$, by the definition in Kock is an element of a monad around identity. The two are in a nontrivial bijection described in Kock. This would be perfect.
Sure, that would be good.
OK, I will do that. Maybe, I may wait a little bit (I would like to think a bit more on synthetic geometry in next couple of days, anyway –there are few points which I would like to sort out beforehand).
One of the points there I would like to have your opinion on. You see, in algebraic geometry when one has a smooth reduced scheme, it is given without nilpotents, and the infinitesimal thickening is a product of a construction, using external infinitesimal object. Kock’s books insist on putting $D$ as a part of the real line, but then he needs another version of $D$ for vector space etc. and axioms for each version and axioms for each version. This becomes cumbersome, to introduces several $D$-s in the game. Then Kock remarks in a footnote at the beginning that, unlike the approach in the book, Lawvere was insisting that one should not start with the extended line, what is about what I sense as better as well. How do you prefer to organize ?
On the other hand, indeed, at the level of categories of qcoh sheaves, one has the thickenings as categories from the very beginning. Somehow the category of sheaves has more interesting subobjects than one can see at the ring level. Indeed, if one takes spectra then infinitesimal neighborhood does not contribute with new points, just with new formal functions. In a special case of a neighborhood of a point, this is to say that the formal functions have zero radius of convergence, hence their common locus is just a point.
At Maurer-Cartan form I have added a brief further paragraph stating the general cohesive definition more explicitly (previously it was stated in the context of smooth cohesion, but of course the general definition is the same).
Also added a similar cohesive paragraph to de Rham complex.
Typo in the link: Maurer-Cartan form.
Thanks! Fixed now.
Another, more recent, reference is
This is Theorem C.2 (read with the notation from Theorem C.1) in the Appendix C. I added it to the list of references, after Helgason.
Well, the formula for the differential of the exponential map – according to the wikipedia –
is from Schur 1891. The fact that the expressions for Maurer-Cartan invariant forms are inverses to the expressions for the invariant vector fields is implicit already in the early papers of Lie. But the history is much more intricate I think. Kirillov’s school used the expressions for the invariant vector fields extensively from 1960-s and early 1970s over reals and complexes. That an extension works in formal geometry over any ring containing rationals is implicit in the study of consequences of Hausdorff formula as explained by Durov who basically uses only the facts from Bourbaki.
P.S. Lie used Hausdorff series and Maurer-Cartan form in this context to prove a formal version of the Lie integration (Lie III), to formal Lie groups. Convergence results for the integration to Lie groups are proved by later people.
So maybe one should dig out the paper where Lie proves the formal version of Lie III theorem, it should be there as in secondary sources which describe his method the Maurer-Cartan and Hausdorff in its connection are used.
But I am not sure.
You can read German so you can also see what has been done by Friedrich (not Issai Schur, cf. Wikipedia: Friedrich Schur) Schur’s cited article on the differential of the exponential map.
Formula (12) is the differential equation we studied at length (and which comes from the Lie’s paper to Meyer in 1874 or so as explained in Bourbaki’s historical essay); the inverse matrix satisfies the Maurer-Cartan.
Yes, (16) is Maurer-Cartan equation, (19) is the equation for the inverse matrix. Schur computes the formula for a solution of (19) hence the formula for the inverse is obvious. Thus, Schur has definitely had it, but it is not clear how he interpreted it. cartan’s reference is 1904 but in more general context.
I see that F. Schur’s (16) is the MC equation (and I’ll add Schur as a reference to that entry for this purpose).
But does Schur really give the coordinate-formula for its pullback under $exp$? Where?
(Of course, in the end, all these formulas follow logically from the foundations of Lie theory. But I would like to cite the actual formula.)
Formula (36) is the formula for the pullback (if you take into account his notation for his $C$-tensors). So this is the formula only up to notational difference from the way you wrote it. He abbreviated the tensor expression with these symbols.
The sign difference is left versus right MC form.
Ah, right. Thanks. That’s remarkable.
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