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added a stubby
to the entry Lie infinity-groupoid.
The punchline is that if we pick a groupal model for – our favorite one is the Lie 2-group – then by the general nonsense of Maurer-Cartan forms on -Lie groups there is a Maurer-Cartan form on . This is, I claim, the universal Ehresmann connection on .
The key steps are indicated in the section now, but not exposed nicely. I expect this is pretty unreadable for the moment and I tried to mark it clearly as being “under construction”. But tomorrow I hope to polish it .
okay, I have been working on it a bit. Not yet fully finalized, but closer to readability. And it’s becoming an opus in several subsections
This looks potentially very interesting, but I feel I am nowhere close to being able to follow it. I would like to be able to help myself by possibly adding some exposition to various articles on which it depends.
For starters, the concept of Chevalley-Eilenberg algebra of an ordinary Lie algebra hasn’t been completely described in the nLab. It would be nice to give a plain meat-and-potatoes definition that a first-year grad student can follow without having to plow through an nPOV explanation.
I may be delayed in such a side project by the fact that starting Friday, I will be in Europe for three weeks with my family, probably without any access to the internet. I will make a brief announcement to this effect elsewhere, just to let people know.
I just discovered that I hadn’t looked at the entry Chevalley-Eilenberg algebra in a long while and completely forgotten in which miserable stubby state it is.
i was under the impression that I had typed out the details already more often than is good for me, but now I discovber that I did that in a bunch of places, but not at that entry.
Details on CE-algebras are instead for instance at Weil algebra (which includes the CE algebra as a piece) and L-infinity-algebra. And I think elsewhere.
I don’t think you spell out the details of CE-algebra at Weil algebra in an easily understandable non-circular way. The details are embedded within L-infinity-algebra, but I didn’t think to look there before you mentioned it just now.
As a general criticism: I would like to see more in the way of bottom-up, not top-down definitions of such basic concepts as CE-algebra. But maybe I’ll stop complaining and start writing. :-)
Okay, I just wrote in a definition at Chevalley-Eilenberg algebra. With a slightly sheepish grin because it makes clear that “bottom-up” is in the eye of the beholder. Please feel free to say more to suit individual taste for “bottom-up”.
Anyway, it’s a definition that I can readily understand and use.
I have expanded this a bit: the “modern” definition of any cohomology since Cartan-Eilenberg’s book is never in tersm of resolution, but rather as a derived functor. So I gave the Ext-definition and claim roughly that the CE as a chain complex provides a resolution.
But now I am a bit unhappy with the definition below. I mean the CE definition works for a -Lie algebra over a commutative unital ring provided is free as -module. In particular, it is not usual to require finite-dimensionality of . The main book on Lie algebra cohomology by Fuks (Fuchs) is mainly dedicated to infinite-dimensional examples; things like Lie algebras of vector fields, current algebras, Virasoro Lie algebra. Finite-dimensionality is important is some closely related issues though.
Well, okayyyy… but somehow this rewriting of what I just wrote seems to have masked over the simple conceptual ideas I tried to introduce in the first place, that
(1) The Grassmann algebra is the free commutative graded algebra in this case, and
(2) Linear maps extend uniquely to derivations on the free (commutative) algebra object.
Urs has erased the whole section which I have added:
The cohomology of a -Lie algebra with coefficients in the left -module is defined as where is the ground field understood as a trivial module over the universal enveloping algebra ; therefore it is a derived functor. Before this was understood in Cartan-Eilenberg’s Homological algebra, Lie algebra cohomology and homology were defined by Chevalley-Eilenberg with a help of concrete Koszul-type resolution which is in this case a chain complex where the first argument is naturally equipped with a differential to start with (see below). This first argument underlines in fact a differential graded commutative algebra called the Chevalley-Eilenberg algebra, which is in lab denoted (as well as its higher categorical analogues) by .
but somehow this rewriting
I changed just one sentence, and added a paragraph. Just the paragraph above, where the last sentence is your sentence mutilated. The rest is done by Urs. Let us together make a version having all the advantages of all points of view.
Oops, I only see this here now. The locks didn’t work. I was writing an entry while you two were, too. Sorry, I didn’t mean to overwrite yours, didn’t even see it.
Yes, I suppose there is something more what was written by Todd what was erased by us in the interlock. Is it recoverable ? My erased stuff was in version 13 and I cut it and pasted it above. It may be moved into the section talking Lie algebra cohomology/cocycles below.
Zoran, sorry for overwriting your paragraph accidentally, but I don’t think it should go at the beginning of the entry on CE-algebras. Instead I think you should move that to Lie algebra cohomology. Notably because you are not in fact talking of the CE-algebra, but of the universal enveloping algebra of .
Todd, okay, we wrote pretty much the same intro paragraph. Currently mine is visible. How do we decide what to do now? Could you maybe just have a look at the entry in its current form (I added a lot more than just the definition) and see where you want to put your paragraph over my material, or in between.
Sorry again for the trouble.
Yes, no material is lost, it’s all visible in the History menu.
But, Urs, there is a section called Lie algebra cohomology within Chevalley-Eilenberg algebra entry. Currently it is pretty complicated, what I wrote is down to earth to start with. Then the cocycle discussion can go.
Yes, no material is lost, it’s all visible in the History menu.
Don’t be sure. There is some version due Todd. It might be that Todd added more stuff when I entered the scene overwriting him.
But, Urs, there is a section called Lie algebra cohomology within Chevalley-Eilenberg algebra entry.
That’s the example how the Lie algebra cohomology is expressed in terms of CE-algebras. What’s complicated about it?
If you don’t want to put your paragraph into Lie algebra cohomology I suggest to put it into universal enveloping algebra.
Notably because you are not in fact talking of the CE-algebra, but of the universal enveloping algebra of
No I am talking of CE algebra. I just wrote it in tersm of as it was meant as a motivating statement. Just the motivating appearance of CE algebra which is discussed in my paragraph is as an ingredient in defining a resolution for Lie algebra cohomology which is the same as the Ext group over the corresponding enveloping algebra.
@Urs #13: I’ll have to look into this later; right now I have to do some play-time with my kids. :-)
What’s complicated about it?
It talks about morphisms of L-infinity algebras. The first 1-categorical introduction must be down to Earth. We have ext-group, usual derived functor. There is then a special resolution. Voila CE algebra. Then higher categorical interpretation…this way it starts with common knowledge and introduces then in higher world.
I have integrated material from above pragraph into Lie algebra cohomology entry (which was before dedicated just to the case of trivial coefficients!). A very short summary
Lie algebra cohomology of a -Lie algebra with coefficients in the left -module is defined as . It can be computed as .
is added at the beginning of the section Lie algebra cohomology within Chevalley-Eilenberg algebra. This treats more general coefficients and makes it accessible to mainstream public. Within the latter entry I think we should (I could) add a parallel paragraph of the role of CE in Lie algebra homology, for which I just created an entry.
Wait a second. There is some genuine confusion here. The Lie algebra cohomology is the usual abelian cohomology (ext functor) in certain abelian category, namely the category of -modules or, what is the same, -modules. The second point of view has the advantage that we know more about the ext groups in categories of modules over associative algebras than arbitrary abelian categories, in particular it motivates Koszul type resolution.
Now, the CE complex in standard homological algebra literature (e.g. modern Weibel’s book) is the whole expression in that category, and that is as in the original CE paper. Of course, the whole thing tensors with (for homology) or homs into (for cohomology) the coefficients – the tensor or hom is over , so one can effectively in the category of -modules (what is not the correct abelian category here) see the smaller module without factor. Of course, one needs to start with otherwise one can not compute the differential correctly as the contributes. However, if you have all the time fixed coefficients, e.g. trivial, then you compute once forever and forget that . Now Urs has made conventions in nlab to never mention . This made my contribution incorrect from that point of view. However we need to state that in the standard literature is taken as part of CE complex, and one needs that part to compute abelian Lie algebra for arbitrary coefficients along the lines I explained.
I do not know how to resolve that issue now.
I htink I roughly resolved it, though have to write more some other day to make it clean. I created an entry Chevalley-Eilenberg chain complex different from Chevalley-Eilenberg cochain complex. Regarding that the chain complex gets into the hom, one can not cheat and shorten it; one needs the full classical thing with . Once one homs it is not that important what conventions are – the full result can be written in two ways, but they are equivalent computationally, though interpreted differently. Of course one can talk the version with coefficients as well as , but that is ok. I hope Urs agrees (that the chain version must have part included while for cochain version it is just matter of writing).
Thanks, Zoran, I’ll have a look when I have a minute.
It is true that I mostly wrote about constant coefficients, being a bit lazy. But I had the section about BRST in there, which is effectively the CE complex with general module coefficients.
Please: CE cochain complex. I agree with you that can drop out in the final step always, but it does affect how the module is mixed into the differential. I will write this remark with more detail. Cf. the calculation at Lie algebra homology. Notice: there is no differential on so that
holds at the level of chain complexes simultaneously for all (while it is OK as graded -spaces).
Similarly
holds, but
as cochain complexes (while it is OK as graded -spaces). In this way is essential.
Urs, in the case that is not finite-dimensional over a field, the Chevalley-Eilenberg cochain complex with trivial coefficients is still , but it is not expressable as . Would you still call it Chevalley-Eilenberg cochain complex (it should be called that way as it is from the original paper and it does the thing with the cohomology) ? Do you still call it Chevalley-Eilenberg algebra ?
Let us furthermore, analyse the duality, even with care of non-finite dimensional situation. In the entry Chevalley-Eilenberg cochain complex you take the finite-dimensional point of view that the Lie algebra generates the cofree cocommutative coalgebra with coproduct and you dualize that. In my understanding this coalgebra is much bigger than its small part which is strictly used in the vector space duality, namely the which is I guess a quotient (?) of . Strictly speaking as a vector space
Right ?
I think it’s still called CE-algebra then. The substitute for in this case is in the original literature taken to be (alternating -multilinear functionals).
I tend to try to ignore this case, because for instane for that no longer dually the model structure on dg-algebras applies, so I don’t quite know what the homotopical category / -category of -algebras of non-finite type is. Though I guess one could use the Getzler-Jones model for differential coalgebras, and often I am thinking that i should be using this much more (meaning: at all :-).
But there is so much more known about dg-algebras of finite type, that I am always being lured in that direction. That allows me to use all rational homotopy reasoning for handling my -algebras, etc. So I am emotionally attached to that case. But more generally one should probably do something else.
I updated the quetsion above, there is a second question there, please look.
I understand that it is harder to have the general case, but for various things in mathematical physics like CFT and BRST exactly the cohomology of infinite-dimensional Lie algebras like (Virasoro, Lie algebras of vector fields, current algebras) is widely used, see e.g. the book by Fuchs etc. At the level of abelian categories, the cohomology is still the usual abelian cohomology, that is the classical point of view which I was entering yesterday. So why to care about the model category structure on dg-algebras if already the abelian homological algebra works nicely with any reinterpretation you like. Now I do not know what changes if we go to general -algebra case, but the point of Lefevre-Hasegawa thesis (pdf) under Bernhard Keller was that the finiteness conditions are artificial an dthat instead for the Koszul duality (see twisting cochain) one works with Maurer-Cartan kind of formalism with cocomplete dg-coalgabras and complete dg-algebras and one has a derived category on one side to describe the homological algebra while one has the Keller-Lefevre-Hasegawa coderived category on the other side.
Of course, the rational homotopy theory means the restriction to field of characteristic zero and so on; but the cohomology of Lie algebras does not need to go into that special case to have nice general and higher categorical meaning, and many interesting homotopical and geometric constructions.
So why to care about the model category structure on dg-algebras if already the abelian homological algebra works nicely with any reinterpretation you like
So what would be the model category of -algebras that you are thinking of?
concerning versus : as vector spaces both are the quotient of a tensor algebra by the skew-symmetrization. So in the finite dim case these are isomorphic as vector spaces, under dualization.
You may need a model category on -algebras to have the nonabelian cohomology, but for abelian cohomology, which we are talking about via Chevalley-Eilenberg stuff, one needs to have homological algebra on the derived category of -modules of a single -algebra; no major difference from the theory of derived category of -modules of a single -algebra which is the triangulated category obtained with the localization with respect to quasiisomorphisms. Model theoretic approach to this kind of categories is in Goncalo Tabuada’s thesis I believe.
Okay, I see, the cat of modues over a single -algebra is sufficient for getting its abelian cohomology, right.
I wait for an aswer to the second question in 25.
That’s the second time you do this to me, right after I did post an answer
as vector spaces both and are the quotient of a tensor algebra by the skew-symmetrization. So in the finite dim case these are isomorphic as vector spaces, under dualization.
concerning versus : as vector spaces both are the quotient of a tensor algebra by the skew-symmetrization. So in the finite dim case these are isomorphic as vector spaces, under dualization.
So you are saying that ?
That’s the second time you do this to me, right after I did post an answer
This is not the answer I asked for. I was asking about the relation with relation to . But as I ask above, you are saying that ? The RHS is always true ingredient for Lie algebra cohomology (which is also ), before the dualization. The LHS is what you are writing for the dual. I want to understand the things. If I have two approaches I need to see their relationship.
As vector spaces and in characteristic 0 , we have , yes.
In particular, I suppose that in general the is not a coalgebra, while the approach to cohomology via still works. It might be that is still some quotient of and that the two are equal in the case of finite-dimensionality.
There is a lightning review of this from page 26 here on.
I suppose that in general the is not a coalgebra.
For me the notation means the free graded co-commutative coalgebra on the vector space .
A problem with large coalgebras is that they do not capture infinite-dimensional things (unless one has topological coalgebras when one can have a coproduct with values in a completed tensor product). In particular, a coalgebra has filtration by finite-dimensional subcoalgebras (a fundamental theorem on coalgebras) unlike the algebras. Thus it is unlikely that coalgebra will say the right thing when replacing some sortof infinite-dimensional algebra. Some similar problems in the comodule theory are resolved by taking the theory of contramodules. This may give a hint why the foundations for semiinfinite cohomology need a homological algebra of the kind Positselski has created. I mean this is just my intuitive guess.
So you are saying we still have a coalgebra. We also still have a Lie algebra to start with. The coalgebra will induce the algebra structure on the dual (I see no need to have finiteness condition for that). Hence we still have the dg algebra structure on the CE cochain complex even without finiteness condition. But the information on Lie bracket is not complete in the CE algebra in this generality. So what is then wrong ? I still do not get it.
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